Keywords: Equations elliptiques non-linéaires, gradient carré, croissance critique, p-Laplacien, Laplacien, Laplacien fractionnaire, gradient Non-linear elliptic equations, gradient square, critical growth, p-Laplacian, Laplacian, franctional Laplacian, non-local gradient. vii

In this thesis, we provide existence, non-existence, uniqueness and multiplicity results for partial differential equations with critical growth in the gradient. The principal techniques employed in our proofs are variational techniques, lower and upper solution theory, a priori estimates and bifurcation theory.

The thesis consists of six chapters. In chapter 0, we introduce the topic of the thesis and we present the main results. Chapter 1 deals with a p-Laplacian type equation with critical growth in the gradient. This equation will depend on a real parameter. Depending on the interval where this parameter lives, we obtain the existence and uniqueness of one solution or we prove the existence and multiplicity of solutions. In chapters 2 and 3, we continue our study in the case where the operator is the Laplacian. However, unlike chapter 1, we study the case where the coefficient functions may change sign. We obtain again existence and multiplicity results. In chapter 4, we study non-local problems of fractional Laplacian type with different non-local gradient terms. We prove existence and non-existence results for different equations of this type. Finally, in chapter 5, we present some open problems related to the content of the thesis and some research perspectives.

También me gustaría dar las GRACIAS al Laboratoire de Mathématiques de Besanc ¸on y al Laboratoire de Mathématiques et applications de Valenciennes por su hospitalidad. Especial menci ón para mis compa ñeros de doctorado en ambos laboratorios. GRACIAS a todas y todos. Espero os vaya todo muy bien y que la vida nos vuelva a reunir en alg ún lugar.

Last but not least: mi mundo "no matemático".

En primer lugar quiero dar las GRACIAS a mis padres por estar siempre ahí. GRACIAS por escucharme incluso cuando no me entendían. No he hablado tanto por teléfono en mi vida. Creo que por culpa de la distancia os valoro todavía más. GRACIAS de todo coraz ón. Jamás habría llegado a este punto sin vosotros. Sé que no siempre ha sido fácil conmigo. Sois enormes. Lo mismo para mis hermanas Ana y Ro. Bueno. . . y Lola y Maica. GRACIAS por estar siempre ahí y hacérmelo todo tan fácil. Finalmente, GRACIAS yaya, Carmen, Agus y Miguel. Mi vida no sería lo mismo sin vosotros. GRACIAS a todos por hacerme sentir tan querido.

También tengo que dar GRACIAS a todos y cada uno de mis amigas y amigos. A mi gente del pueblo: GRACIAS por hacerme tan fácil todos y cada uno de mis "regresos". Es como si el tiempo no pasase. GRACIAS Lula, Juana, María, Jose, Pepe, Juan Fran, Efe y Alfonso. A mis amigos de la UM-UGR: GRACIAS por estar siempre ahí, en las buenas y las no tan buenas. En particular, GRACIAS a Catiana, Gimmy, Guille y David. Finalmente, GRACIAS a mi familia "Bisontine" por haber hecho de Besanc ¸on mi segundo hogar. GRACIAS Julia, Melina, Melissa, Mathilde, Alicia y Andrea (garc ¸on) por hacerme sentir en casa.

GRACIAS a toda la gente que me ha aportado algo durante estos tres a ños. Ya sea una bonita conversaci ón, una sonrisa, una triste discusi ón, un apoyo cuando me hacía falta o un bello paseo. GRACIAS a toda mi bella gente de Aachen, Besanc ¸on y Valenciennes.

¡GRACIAS A TODAS Y TODOS!

Résumé

Dans cette thèse, nous donnons des résultats d'existence, de non-existence, d'unicité et de multiplicité de solutions pour des équation aux dérivées partielles avec croissance critique dans le gradient. Les principales méthodes utilisées dans nos preuves sont des arguments variationnels, la théorie des sous et sur-solutions, des estimations à priori et la théorie de la bifurcation.

La thèse se compose de six chapitres. Dans le chapitre 0 nous introduisons le sujet de thèse et nous présentons les résultats principaux. Le chapitre 1 porte sur l'étude d'une équation du type p-Laplacien avec croissance critique dans le gradient et dépendant d'un paramètre. En fonction de l'intervalle o ù se trouve le paramètre, nous obtenons l'existence et l'unicité d'une solution ou nous montrons l'existence et la multiplicité de solutions. Dans les chapitres 2 et 3, nous poursuivons notre étude dans le cas o ù l'opérateur utilisé est le Laplacien mais, contrairement au chapitre 1, nous étudions le cas o ù les coefficients changent de signe. Nous obtenons à nouveau des résultats d'existence et de multiplicité de solutions. Dans le chapitre 4, nous étudions des problèmes nonlocaux du type Laplacien fractionnaire avec différents termes de gradient non-local. Nous montrons des résultats d'existence et de non-existence de solutions pour différentes équations de ce type. Finalement, dans le chapitre 5 nous présentons quelques problèmes ouverts liés au contenu de la thèse et des perspectives de recherche.

Introduction

This thesis is devoted to the study of elliptic partial differential equations involving nonlinear gradient terms. More precisely, we investigate the existence, non-existence and multiplicity of solutions to several problems with critical growth in the gradient. As a model case, we can think of the boundary value problem

-∆u = c(x)u + µ(x)|∇u| 2 + h(x),
in Ω, u = 0, on ∂Ω, [START_REF] Hamid | On the connection between two quasilinear elliptic problems with source terms of order 0 or 1[END_REF] where Ω ⊂ R N is a bounded domain with smooth boundary, c and h belong to suitable L p -spaces, µ belongs to L ∞ (Ω) and the solutions are searched in H 1 0 (Ω) ∩ L ∞ (Ω). There exist several mathematical reasons that make the study of nonlinear elliptic PDE with quadratic growth in the gradient interesting. For instance, J. L. Kazdan and R. J. Kramer observed in 1978 that second order PDE with quadratic growth in the gradient are invariant under changes of variable of type v = F(u). This took them to claim on [78, page 619] that "In the long run, the class of semilinear equations should be less important than some more general class of equations that is invariant under changes of variables". From a pure mathematical point of view, it is also worth noting that, in Riemannian geometry, this type of equations naturally appears in the study of gradient Ricci solitons, see for instance [89, Section 1]. On the other hand, concerning more practical reasons, we would like to mention that problem [START_REF] Hamid | On the connection between two quasilinear elliptic problems with source terms of order 0 or 1[END_REF] with c ≡ 0 corresponds to the stationary case of the Kardar-Parisi-Zhang model of growing interfaces introduced in [START_REF] Kardar | Dynamic scaling of growing interfaces[END_REF]. Now, before going further in our introduction, we would like to clarify two points. If one thinks about the classical theory of elliptic PDEs, it seems natural to look for solutions to [START_REF] Hamid | On the connection between two quasilinear elliptic problems with source terms of order 0 or 1[END_REF] in H 1 0 (Ω). Hence, a normal question is:

Why are we looking for solutions in H 1 0 (Ω) ∩ L ∞ (Ω)?

For N ≥ 3, let us consider the boundary boundary problem

-∆u = |∇u| 2 , u ∈ H 1 0 (B 1 (0)), (2)
where B 1 (0) := {x ∈ R N : |x| < 1}. On the one hand, on [4, page 21], it was observed that, for any 0 ≤ m < 1,

u m (x) = log |x| 2-N -m 1 -m ,
is a solution to [START_REF] Abdellaoui | On the fractional p-Laplacian equations with weight and general datum[END_REF]. On the other hand, by [START_REF] Arcoya | Remarks on the uniqueness for quasilinear elliptic equations with quadratic growth conditions[END_REF]Theorem 1.1] we know that the unique solution to (2) belonging to L ∞ (Ω) is the trivial one, namely u ≡ 0. Hence, we easily observe that the set of solutions to (2) is completely different when imposing or not that the solutions belong to L ∞ (Ω).

Another question we should address is:

Why do we say that (1) has critical growth in the gradient?

For N ≥ 3 and q > 1, let us consider the boundary value problem

-∆u = |∇u| q , u ∈ H 1 0 (B 1 (0)) ∩ L ∞ (B 1 (0)), (3)
and distinguish two cases: a) 1 < q ≤ 2: It follows from [START_REF] Arcoya | Remarks on the uniqueness for quasilinear elliptic equations with quadratic growth conditions[END_REF]Theorem 1.1] that the unique solution to (3) is u ≡ 0.

b) q > 2: Let us introduce β = q -2 q -1 and α = (N -2 + β)

1 q-1 β ,
and define v(x) := α(1 -|x| β ). By direct computations we observe that u ≡ 0 and v are both solutions to [START_REF] Abdellaoui | Caffarelli-Kohn-Nirenberg type inequalities of fractional order with applications[END_REF]. Hence, for q > 2, problem (3) has at least two different solutions.

We observe that the set of solutions to (3) is different for 1 < q ≤ 2 and q > 2 and so, that the problems behave very differently. We conclude then that q = 2 is a turning point in the behaviour of the problem. This justifies somehow the terminology of critical growth in the gradient when we refer to the power q = 2.

For 1 < p < ∞, let ∆ p u = div(|∇u| p-2 ∇u) denote the p-Laplacian operator (observe that for p = 2 it is precisely the Laplacian). In the first part of the thesis, Chapters 1, 2 and 3, we deal with the existence and multiplicity of solutions to boundary value problems of the form

-∆ p u = c λ (x)|u| p-2 u + µ(x)|∇u| p + h(x), u ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω), (4)
with c λ depending on a real parameter λ. Here, Ω ⊂ R N , N ≥ 2, is a bounded domain with smooth boundary, c λ and h belong to L q (Ω) for some q > max{N /p, 1} and µ belongs to L ∞ (Ω). Let us specify that in Chapters 2 and 3 we consider p = 2. In that case, problem (4) reduces to

-∆u = c λ (x)u + µ(x)|∇u| 2 + h(x), u ∈ H 1 0 (Ω) ∩ L ∞ (Ω), (5)
which is precisely the model case [START_REF] Hamid | On the connection between two quasilinear elliptic problems with source terms of order 0 or 1[END_REF] with c ≡ c λ .

First of all, we are going to present the state of the art concerning problems (4) and ( 5) when we began the thesis in October 2016.

The study of nonlinear elliptic PDEs with a gradient dependence up to the critical growth was essentially initiated by L. Boccardo, F. Murat and J.-P. Puel in the 80's [START_REF] Boccardo | Existence de solutions faibles pour des équations elliptiques quasilinéaires à croissance quadratique[END_REF][START_REF]Quelques propriétés des opérateurs elliptiques quasi linéaires[END_REF][START_REF]L ∞ estimate for some nonlinear elliptic partial differential equations and application to an existence result[END_REF]. This type of problems have generated since then a large literature. In addition to several works strictly related to the content of this thesis (that we will detail next), several directions have been investigated. For instance, let us mention here some recent works concerning subcritical growth in the gradient [START_REF] Grenon | Existence and a priori estimate for elliptic problems with subquadratic gradient dependent terms[END_REF][START_REF]A priori estimates and existence for elliptic equations with gradient dependent terms[END_REF][START_REF]The "ergodic limit" for a viscous Hamilton-Jacobi equation with Dirichlet conditions[END_REF], supercritical growth in the gradient [START_REF] Phuc | Morrey global bounds and quasilinear Riccati type equations below the natural exponent[END_REF], low regularity coefficients and regularizing effects [START_REF] Arcoya | Regularizing effect of the interplay between coefficients in some elliptic equations[END_REF][START_REF]Regularizing effect of L q interplay between coefficients in some elliptic equations[END_REF] and pointwise estimates via symmetrization methods [START_REF] Hamel | Comparison results and improved quantified inequalities for semilinear elliptic equations[END_REF]. Now, we focus precisely on the literature concerning existence and multiplicity of solutions to problems (4) and [START_REF] Abdellaoui | Nonlinear fractional Laplacian problems with nonlocal "gradient terms[END_REF].

In the case where c λ (x) ≤ -α 0 < 0 a.e. in Ω for some α 0 > 0, now referred to as the coercive case, the existence of a solution to (4) is a particular case of the results of [START_REF] Boccardo | Existence de solutions faibles pour des équations elliptiques quasilinéaires à croissance quadratique[END_REF][START_REF]Quelques propriétés des opérateurs elliptiques quasi linéaires[END_REF][START_REF]L ∞ estimate for some nonlinear elliptic partial differential equations and application to an existence result[END_REF]. Nevertheless, it is still an open question if the uniqueness of such solution holds. Some partial results are in order: for p = 2, the uniqueness follows from the results of [START_REF] Barles | Remarks on the maximum principle for nonlinear elliptic PDEs with quadratic growth conditions[END_REF][START_REF] Barles | Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions[END_REF]; for 1 < p < 2, it seems possible to prove the uniqueness of solutions modifying the proof of [START_REF] Arcoya | Remarks on the uniqueness for quasilinear elliptic equations with quadratic growth conditions[END_REF]Theorem 1.1] and, for 2 < p < ∞ and more regular coefficients (c λ , µ, h ∈ L ∞ (Ω)), it seems reachable adapting the proof of [START_REF] Leonori | Comparison principles for p-Laplace equations with lower order terms[END_REF]Theorem 1.2]. The weakly coercive case c λ ≡ 0 was studied in [START_REF] Ferone | Nonlinear problems having natural growth in the gradient: an existence result when the source terms are small[END_REF] where, for µh N /p small enough, the existence of a solution to (4) is obtained. For p = 2, the uniqueness of solution was also obtained in [START_REF] Barles | Remarks on the maximum principle for nonlinear elliptic PDEs with quadratic growth conditions[END_REF] requiring h small enough in an appropriate sense. The limit coercive case where one only requires c λ ≤ 0 a.e. in Ω (i.e. allowing parts of the domain where c λ ≡ 0 and parts of it where c λ < 0) proved to be more complex to treat. Until this thesis, just the situation where p = 2 had been treated. In the recent paper [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF], the authors observed that the existence of a solution to [START_REF] Abdellaoui | Nonlinear fractional Laplacian problems with nonlocal "gradient terms[END_REF] is not guaranteed and gave sufficient conditions to ensure such existence. In that paper, the uniqueness of solutions is also proved in this framework. We refer likewise to [START_REF] Arcoya | Remarks on the uniqueness for quasilinear elliptic equations with quadratic growth conditions[END_REF] where more general uniqueness results were obtained. Nevertheless, as already mentioned, the papers concerning the limit coercive case c λ ≤ 0 a.e. in Ω dealt with [START_REF] Abdellaoui | Nonlinear fractional Laplacian problems with nonlocal "gradient terms[END_REF], i.e. p = 2. Finally, let us stress that all these results were obtained without requiring any sign conditions on µ and h.

As we will detail later (see Section 0.1), if µ is a constant, the change of variable v = p-1 µ (e µ p-1 u -1) allows to transform problem (4) into a new one which admits a variational formulation (i.e. a problem whose solutions can be obtained as critical points of an associated functional). When c λ ≤ -α 0 < 0 a.e. in Ω, the associated functional, defined in W 1,p 0 (Ω), is coercive. If c λ ≤ 0 a.e. in Ω, the coerciveness may be lost but it is possible to find sufficient conditions to ensure it. Nevertheless, as soon as c + λ 0, the functional becomes unbounded from below. Hence, if c λ 0 a.e. in Ω, i.e. c λ 0 or c λ changes sign, problem (4) behaves very differently and becomes much richer than for c λ ≤ 0 a.e. in Ω. For c λ 0 a.e. in Ω, up to Chapter 1 of the present thesis, just the particular choice p = 2 had been addressed. Furthermore, even for p = 2, this case remained unexplored until very recently.

Let us fix for the moment p = 2. The first paper dealing with c λ 0 a.e. in Ω was [START_REF] Jeanjean | Existence and multiplicity for elliptic problems with quadratic growth in the gradient[END_REF] where the authors considered c λ = λc 0. Following [START_REF] Sirakov | Solvability of uniformly elliptic fully nonlinear PDE[END_REF], which addressed a particular case, the authors studied [START_REF] Abdellaoui | Nonlinear fractional Laplacian problems with nonlocal "gradient terms[END_REF] with µ(x) ≡ µ > 0 but without sign conditions on h. For λ > 0 and µh N /2 small enough it was proved the existence of at least two solutions to (5) using the previously mentioned variational formulation. The main issue to derive the existence of solutions to [START_REF] Abdellaoui | Nonlinear fractional Laplacian problems with nonlocal "gradient terms[END_REF] was then to prove the boundedness of the Palais-Smale sequences to the variational problem and to ensure that the change of variable could be undone (i.e. to verify that the solutions v to the variational problem satisfied v > -1/µ). This result has now been complemented in several ways. The restriction µ constant was first removed in [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF] and extended to µ(x) ≥ µ 1 > 0 a.e. in Ω, at the expense of adding the hypothesis h 0. In this direction we refer also to [START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF], where, assuming stronger regularity on c and h, the authors removed the condition h 0. In [START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF], under different sets of assumptions, the authors clarified the structure of the set of solutions to [START_REF] Abdellaoui | Nonlinear fractional Laplacian problems with nonlocal "gradient terms[END_REF] in the case λc 0. Note that, in the frame of viscosity solutions and fully nonlinear equations and under corresponding assumptions, similar conclusions were obtained very recently in [START_REF] Nornberg | A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient[END_REF]. In [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF][START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF][START_REF] Nornberg | A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient[END_REF] where µ is non constant, the variational approach used in [START_REF] Jeanjean | Existence and multiplicity for elliptic problems with quadratic growth in the gradient[END_REF] was no longer available. Topological arguments, relying on the derivation of a priori bounds for certain classes of solutions, were used.

Before we developed Chapters 2 and 3 of the present thesis, the only known work where c λ may change sign was [START_REF] Jeanjean | Multiple solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF] (see also [START_REF] García-Meli Án | A priori bounds and existence of solutions for slightly superlinear elliptic problems[END_REF] for related problems). Assuming µ > 0 constant, h 0 and µh and c + λ small in an appropriate sense, the existence of at least two non-negative solutions was proved. Since µ is constant in [START_REF] Jeanjean | Multiple solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF], the problem fitted in the variational framework set up in [START_REF] Jeanjean | Existence and multiplicity for elliptic problems with quadratic growth in the gradient[END_REF]. The main issue was then to prove that the Palais-Smale sequences were bounded. Due to the indefinite sign of c λ , several difficulties had to be faced.

When c λ 0 a.e. in Ω (i.e. the non-coercive case), all the above mentioned results require either µ to be constant or to be uniformly bounded from below by a positive constant. In [START_REF] Souplet | A priori estimates and bifurcation of solutions for an elliptic equation with semidefinite critical growth in the gradient[END_REF], assuming that c λ , µ and h were non-negative, a first attempt to remove these restrictions on µ was presented. Following the approach of [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF], the proofs of the existence results reduce to obtaining a priori bounds on the non negative solutions to [START_REF] Abdellaoui | Nonlinear fractional Laplacian problems with nonlocal "gradient terms[END_REF]. The approach developed in [START_REF] Souplet | A priori estimates and bifurcation of solutions for an elliptic equation with semidefinite critical growth in the gradient[END_REF] is based on interpolation and elliptic estimates in weighted Lebesgue spaces. It works well in low dimension but the possibility to extend it to an arbitrary dimension is not apparent.

In the rest of the introduction we are going to present the main contributions of this thesis. We split it into four sections. Each section corresponds to one chapter of the thesis. In Section 0.1, which corresponds to Chapter 1, we deal with (4) assuming that c λ has a sign. Sections 0.2 and 0.3, which correspond to Chapters 2 and 3 respectively, are devoted to the study of [START_REF] Abdellaoui | Nonlinear fractional Laplacian problems with nonlocal "gradient terms[END_REF] in the case where c λ may change sign. Finally, in Section 0.4, which corresponds to Chapter 4, we address a different topic. We present there our main contributions to the study of nonlocal problems involving nonlocal "gradient terms". Section 0.4 is self-contained and independent of the others three sections.

Each chapter of the present thesis (except for Chapter 5) corresponds to a research article (already published or preprint) and can be read independently. Chapter 1 corresponds to the paper [START_REF] Coster | Existence and multiplicity for elliptic p-Laplacian problems with critical growth in the gradient[END_REF] in collaboration with C. De Coster. Chapter 2 corresponds to [START_REF] Coster | A priori bounds and multiplicity of solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF] and it is done in collaboration with C. De Coster and L. Jeanjean. Chapter 3 is based on [START_REF]Existence and multiplicity for an elliptic problem with critical growth in the gradient and signchanging coefficients[END_REF] which is a joint work with C. De Coster. Finally, Chapter 4 corresponds to [START_REF] Abdellaoui | Nonlinear fractional Laplacian problems with nonlocal "gradient terms[END_REF] and it is a joint work with B. Abdellaoui. Finally, in Chapter 5, we present some open problems and describe some future projects.

Notation.

1) We denote R + = (0, +∞), R -= (-∞, 0) and N = {1, 2, 3, . . .}.

2) In R N , we use the notations |x| = (x 2 1 + . . . + x 2 N ) 1/2 and B R (y) = {x ∈ R N : |x -y| < R}. 3) For v ∈ L 1 (Ω) we define v + = max(v, 0) and v -= max(-v, 0). 4) For h 1 , h 2 ∈ L 1 (Ω) we write:

• h 1 ≤ h 2 if h 1 (x) ≤ h 2 (x) for a.e. x ∈ Ω,

• h 1 h 2 if h 1 ≤ h 2 and meas({x ∈ Ω : h 1 (x) < h 2 (x)}) > 0.

c λ has a sign

In the first chapter of the thesis, Chapter 1, we assume that c λ has a sign and we address the more general case problem (4). For any 1 < p < ∞, we consider the boundary value problem

-∆ p u = λc(x)|u| p-2 u + µ(x)|∇u| p + h(x), u ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω), (6)
under the assumption

           Ω ⊂ R N , N ≥ 2
, is a bounded domain with ∂Ω of class C 0,1 , c and h belong to L q (Ω) for some q > max{N /p, 1}, c 0 and µ ∈ L ∞ (Ω) .

(

) 7
We prove existence and uniqueness results in the limit coercive case λ ≤ 0 and existence and multiplicity results in the non-coercive case λ > 0. Moreover, considering stronger regularity assumptions on the coefficient functions, we clarify the structure of the set of solutions in the non-coercive case λ > 0. We provide now some details on the main results of Chapter 1.

To state the first main result of this thesis let us define

m + p,λ :=          inf u∈W λ Ω |∇u| p - µ + ∞ p -1 p-1 h(x)|u| p dx, if W λ ∅ , + ∞, if W λ = ∅ ,
and

m - p,λ :=          inf u∈W λ Ω |∇u| p + µ - ∞ p -1 p-1 h(x)|u| p dx, if W λ ∅ , + ∞, if W λ = ∅ ,
where W λ := w ∈ W 1,p 0 (Ω) : λc(x)w(x) = 0 a.e. x ∈ Ω , w W 1,p 0 (Ω) = 1 . The following result, whose proof combines lower and upper solution and variational techniques, generalizes those obtained in [18, Section 3].

Theorem 0.1.1. [Theorem 1.1.1, Chapter 1] Assume that [START_REF]Towards a deterministic KPZ equation with fractional diffusion: the stationary problem[END_REF] holds and that λ ≤ 0. Then if m + p,λ > 0 and m - p,λ > 0, the problem [START_REF] Abdellaoui | Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential[END_REF] has at least one solution.

Remark 0.1.1.

a) The space W λ is independent of the size of λ. It depends only on the fact that λ = 0 or λ 0.

b) The assumption m + p,λ > 0 and m - p,λ > 0 connects the cases λc(x) ≤ -α 0 < 0 and λc(x) ≡ 0. On one hand, if λc(x) < 0 a.e. in Ω, we have W λ = ∅ and hence m + p,λ = m - p,λ = +∞. On the other hand, if λc(x) ≡ 0, then W λ = W In [START_REF] Arcoya | Remarks on the uniqueness for quasilinear elliptic equations with quadratic growth conditions[END_REF], for p = 2 but assuming only [START_REF]Towards a deterministic KPZ equation with fractional diffusion: the stationary problem[END_REF], the uniqueness of solutions when λ ≤ 0 was obtained as a direct consequence of a suitable comparison principle, see [START_REF] Arcoya | Remarks on the uniqueness for quasilinear elliptic equations with quadratic growth conditions[END_REF]Corollary 3.1]. Nevertheless, as we show in Remark 1.3.3, such kind of principle does not hold in general when p 2. Actually, the issue of uniqueness for equations of the form of ( 6) appears widely open. If partial results seem reachable adapting existing techniques (see for instance [START_REF] Damascelli | Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-Laplace equations[END_REF][START_REF] Leonori | Comparison principles for p-Laplace equations with lower order terms[END_REF][START_REF] Porretta | On the comparison principle for p-Laplace type operators with first order terms[END_REF][START_REF] Pucci | The maximum principle[END_REF]), a result covering the full generality of (6) seems, so far, out of reach. Theorem 0.1.2 below, whose proof makes use of some ideas from [START_REF] Abdellaoui | Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential[END_REF], crucially relies on the assumption that µ is constant. It permits however to treat the limit coercive case where c may vanish in some parts of Ω.

Theorem 0.1.2. [Theorem 1.1.2, Chapter 1] Assume that [START_REF] Abdellaoui | Elliptic problems with a Hardy potential and critical growth in the gradient: non-resonance and blow-up results[END_REF] holds and suppose λ ≤ 0. Then [START_REF] Abdellaoui | Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential[END_REF] has at most one solution.

Actually, Theorem 0.1.2 above is a particular case of the more general comparison principle that we state next. Let us consider the boundary value problem

-∆ p u = µ|∇u| p + f (x, u) , u ∈ W 1,p 0 (Ω) ∩ C(Ω) , ( 9
)
under the assumption 1 -Carathéodory and f (x, s) ≤ f (x, t) for a.e. x ∈ Ω and all t ≤ s, µ > 0. b) The assumption µ > 0 is not a restriction. If u ∈ W 1,p 0 (Ω) ∩ C(Ω) is a solution to [START_REF] Adams | Sobolev spaces[END_REF] with µ < 0 then w = -u ∈ W 1,p 0 (Ω) ∩ C(Ω) is a solution to -∆ p w = -µ|∇w| pf (x, -w) , w ∈ W 1,p 0 (Ω) ∩ C(Ω) , with -f (x, -s) satisfying the assumption [START_REF] Alama | On semilinear elliptic equations with indefinite nonlinearities[END_REF].

           Ω ⊂ R N , N ≥ 2 , is a bounded domain with ∂Ω of class C 0,1 , f : Ω × R → R is L
Theorem 0.1.3. [Theorem 1.3.1, Chapter 1] Assume that [START_REF] Alama | On semilinear elliptic equations with indefinite nonlinearities[END_REF] holds. If u 1 , u 2 ∈ W 1,p (Ω) ∩ C(Ω) are respectively a lower and an upper solution to [START_REF] Adams | Sobolev spaces[END_REF], then u 1 ≤ u 2 .

Remark 0.1.3. The definition of lower and upper solution to (9) that we use is stated in Definition 1.2.1.

Let us highlight a particular case of (6) that will play an important role in the subsequent results. Namely, we introduce the problem

-∆ p u = µ|∇u| p + h(x), u ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω). ( 11
)
First of all, using the notation

m p := inf Ω |∇w| p - µ p -1 p-1 h(x)|w| p dx : w ∈ W 1,p 0 (Ω), w W 1,p 0 (Ω) = 1 , (12)
we completely characterise the existence of solution to [START_REF] Allegretto | A Picone's identity for the p-Laplacian and applications[END_REF].

Theorem 0.1.4. [Theorem 1.1.3, Chapter 1] Assume that [START_REF] Abdellaoui | Elliptic problems with a Hardy potential and critical growth in the gradient: non-resonance and blow-up results[END_REF] holds. Then [START_REF] Allegretto | A Picone's identity for the p-Laplacian and applications[END_REF] has a solution if, and only if, m p > 0.

Remark 0.1.4. This result again improves, for µ constant, [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF] and it allows to observe that, in case h ≤ 0, [START_REF] Allegretto | A Picone's identity for the p-Laplacian and applications[END_REF] has always a solution while the case h 0 is the "worse" case for the existence of a solution. In case h changes sign, the negative part of h "helps" in order to have a solution to [START_REF] Allegretto | A Picone's identity for the p-Laplacian and applications[END_REF].

Gluing together Theorems 0.1.1, 0.1.2 and 0.1.4 we obtain the following existence and uniqueness result for λ ≤ 0.

Corollary 0.1.5. [Corollary 1.1.4, Chapter 1] Assume that [START_REF] Abdellaoui | Elliptic problems with a Hardy potential and critical growth in the gradient: non-resonance and blow-up results[END_REF] holds and suppose that [START_REF] Allegretto | A Picone's identity for the p-Laplacian and applications[END_REF] has a solution. Then, for all λ ≤ 0, [START_REF] Abdellaoui | Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential[END_REF] has an unique solution.

Having at hand this information for the limit coercive case λ ≤ 0, we turn to the study of the non-coercive case, namely when λ > 0. First, using mainly variational techniques as in [START_REF] Jeanjean | Existence and multiplicity for elliptic problems with quadratic growth in the gradient[END_REF], we prove the following result. Theorem 0.1.6. [Theorem 1.1.5, Chapter 1] Assume that [START_REF] Abdellaoui | Elliptic problems with a Hardy potential and critical growth in the gradient: non-resonance and blow-up results[END_REF] holds and suppose that [START_REF] Allegretto | A Picone's identity for the p-Laplacian and applications[END_REF] has a solution. Then there exists Λ > 0 such that, for any 0 < λ < Λ, (6) has at least two solutions.

Next, considering stronger regularity assumptions and combining the theory of lower and upper solution with variational techniques, we derive a more precise information on the structure of the set of solutions in the non-coercive case. These informations complement Theorem 0.1.6. Let us denote by γ 1 > 0 the first eigenvalue of the problem -∆ p u = γc(x)|u| p-2 u , u ∈ W 1,p 0 (Ω). [START_REF] Ambrosetti | Dual variational methods in critical point theory and applications[END_REF] Under the assumption

           Ω ⊂ R N , N ≥ 2
, is a bounded domain with ∂Ω of class C 2 , c and h belong to L ∞ (Ω) , c 0 and µ > 0 , [START_REF] Applebaum | Lévy processes and stochastic calculus[END_REF] we prove the following result.

Theorem 0.1.7. [Theorem 1.1.6, Chapter 1] Assume that [START_REF] Applebaum | Lévy processes and stochastic calculus[END_REF] holds and suppose that [START_REF] Allegretto | A Picone's identity for the p-Laplacian and applications[END_REF] has a solution u 0 . Then:

• If h 0, for every λ > 0, [START_REF] Abdellaoui | Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential[END_REF] has at least two solutions u λ,1 , u λ,2 ∈ C 1 0 (Ω) with u λ,1 0.

• If h 0, then u 0 0 and there exists λ ∈ (0, γ 1 ) such that:

• for every 0 < λ < λ, [START_REF] Abdellaoui | Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential[END_REF] has at least two solutions u λ,1 , u λ,2 ∈ C 1 0 (Ω) satisfying u λ,i ≥ u 0 for i = 1, 2;

• for λ = λ, [START_REF] Abdellaoui | Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential[END_REF] has at least one solution u λ ∈ C 1 0 (Ω) satisfying u λ ≥ u 0 ; • for any λ > λ, [START_REF] Abdellaoui | Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential[END_REF] has no non-negative solution. b) As observed above (see for instance Theorem 0.1.4), in the case h 0, the assumption that [START_REF] Allegretto | A Picone's identity for the p-Laplacian and applications[END_REF] has a solution is automatically satisfied. c) In the case µ < 0, we have the opposite result i.e., two solutions for every λ > 0 in case h 0 and, in case h 0, the existence of λ > 0 such that [START_REF] Abdellaoui | Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential[END_REF] has at least two negative solutions, at least one negative solution or no non-positive solution according to 0 < λ < λ, λ = λ or λ > λ.

In case h 0, we know that for λ > λ (given in the previous theorem), [START_REF] Abdellaoui | Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential[END_REF] has no non-negative solutions but this does not exclude the possibility of having negative or sign changing solutions. Changing the problem a little we are able to prove again the existence of at least two solutions. We consider the boundary value problem -∆ p u = λc(x)|u| p-2 u + µ|∇u| p + kh(x) , u ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) , [START_REF] Arcoya | Regularizing effect of the interplay between coefficients in some elliptic equations[END_REF] with a dependence in the size of h and we obtain the following result.

Theorem 0.1.8. [Theorem 1.1.7, Chapter 1] Assume that [START_REF] Applebaum | Lévy processes and stochastic calculus[END_REF] holds and that h 0. Let

k 0 = sup k ∈ [0, +∞) : ∀ w ∈ W 1,p 0 (Ω) , Ω |∇w| p - µ p -1 p-1 k h(x)|w| p dx > 0 .
Then:

• For all λ ∈ (0, γ 1 ), there exists k = k(λ) ∈ (0, k 0 ) such that, for all k ∈ (0, k), the problem [START_REF] Arcoya | Regularizing effect of the interplay between coefficients in some elliptic equations[END_REF] has at least two solutions u λ,1 , u λ,2 ∈ C 1 0 (Ω) with u λ,i 0 and for all k > k, the problem [START_REF] Arcoya | Regularizing effect of the interplay between coefficients in some elliptic equations[END_REF] has no solution. Moreover, the function k(λ) is non-increasing.

• For λ = γ 1 , the problem [START_REF] Arcoya | Regularizing effect of the interplay between coefficients in some elliptic equations[END_REF] has a solution if and only if k = 0. In that case, the solution is unique and it is equal to 0.

• For all λ > γ 1 , there exist 0 < k1 ≤ k2 < +∞ such that, for all k ∈ (0, k1 ), the problem [START_REF] Arcoya | Regularizing effect of the interplay between coefficients in some elliptic equations[END_REF] has at least two solutions u λ,1 , u λ,2 ∈ C 1 0 (Ω) with u λ,1 0 and min u λ,2 < 0, for all k > k2 , the problem [START_REF] Arcoya | Regularizing effect of the interplay between coefficients in some elliptic equations[END_REF] has no solution and, in case k1 < k2 , for all k ∈ ( k1 , k2 ), the problem [START_REF] Arcoya | Regularizing effect of the interplay between coefficients in some elliptic equations[END_REF] has at least one solution u λ with u λ 0 and min u λ < 0. Moreover, the function k 1 (λ) is non-decreasing. Some comments about the proofs of Theorems 0.1.6, 0.1.7 and 0.1.8 are in order. As already said, when µ > 0 is a constant, it is possible to perform a change of variable and to reduce (6) to a semilinear problem. More precisely, introducing

2 2 1 0 k(λ) γ 1 λ k 0 k k 1 (λ) k 2 (λ)
v = p -1 µ e µ p-1 u -1 ,
one can easily check that u is a solution to (6) if, and only if, v > -

p-1 µ is a solution to -∆ p v = λc(x)g(v) + 1 + µ p -1 v p-1 h(x) , v ∈ W 1,p 0 (Ω) , ( 16
)
where g satisfies

g(s) = p -1 µ 1 + µ p -1 s ln 1 + µ p -1 s p-2 p -1 µ 1 + µ p -1 s ln 1 + µ p -1 s , if s > - p -1 µ . ( 17
)
Working with problem ( 16) has the advantage that, with a suitable choice of g when s ≤ -p-1 µ , the problem has a variational structure. Nevertheless, we have to overcome several difficulties.

First we have to verify that the solutions to ( 16) satisfy v > -p-1 µ . In order to do that, we prove the existence of a non-positive lower solution u λ to [START_REF] Abdellaoui | Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential[END_REF] such that every upper solution β to (6) satisfies β ≥ u λ . This allows us to transform problem ( 16) into a new one, which, unlike the variational problems considered in [START_REF] Jeanjean | Multiple solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF][START_REF] Jeanjean | Existence and multiplicity for elliptic problems with quadratic growth in the gradient[END_REF], is completely equivalent to [START_REF] Abdellaoui | Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential[END_REF]. More precisely, let us define

α λ = p -1 µ e µ p-1 u λ -1 ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) , ( 18
)
and observe that, since u λ is non-positive and belongs to L ∞ (Ω), 0 ≥ α λ ≥ -p-1 µ + ε for some ε > 0. Then, for any λ ∈ R, we consider the auxiliary problem

-∆ p v = f λ (x, v) , v ∈ W 1,p 0 (Ω) , ( 19
)
where

f λ (x, s) =              λc(x)g(s) + 1 + µ p -1 s p-1 h(x) , if s ≥ α λ (x) , λc(x)g(α λ (x)) + 1 + µ p -1 α λ (x) p-1 h(x) , if s ≤ α λ (x) , (20)
and we prove in Proposition 1.5.2 that v ∈ W

u = p -1 µ ln 1 + µ p -1 v ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω)
is a solution to [START_REF] Abdellaoui | Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential[END_REF].

The main advantages of problem [START_REF] Barles | Remarks on the maximum principle for nonlinear elliptic PDEs with quadratic growth conditions[END_REF] are that it admits a variational formulation and it is completely equivalent to [START_REF] Abdellaoui | Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential[END_REF]. We shall then look for solutions to [START_REF] Barles | Remarks on the maximum principle for nonlinear elliptic PDEs with quadratic growth conditions[END_REF] as critical points of an associated functional I λ . When λ > 0, the functional I λ is unbounded from below and presents a concave-convex type geometry. Then, in trying to obtain critical points of I λ , the fact that g is only slightly superlinear at infinity is a difficulty. It implies that I λ does not satisfies an Ambrosetti-Rabinowitz-type condition and proving that the Palais-Smale (Cerami in our case) sequences are bounded becomes challenging. Actually, the proof of Lemma 1.8.1 (boundedness of the Cerami sequences) is one of the main difficulties of Chapter 1 and make use of some ideas that are new in the literature and may be useful in other settings.

Once we have proved the Cerami condition (see Definition 1.2.3) for I λ with λ > 0, in order to prove Theorems 0.1.6, 0.1.7 and 0.1.8, we look for critical points of I λ which are either localminimum or of a mountain-pass type. In Theorem 0.1.6 the geometry of I λ is "simple" and permits to use only variational arguments. In Theorems 0.1.7 and 0.1.8 however it is not so clear, looking directly to I λ , where to search for critical points. We shall then combine variational techniques and lower and upper solution arguments. In both theorems a first solution is obtained through the existence of well-ordered lower and upper solutions. This solution is further proved to be a local minimum of I λ and it is then possible to obtain a second solution by a mountain pass argument.

Problem (5) with the three coefficient functions changing sign

Let us fix from now on p = 2, i.e. we are going to deal with [START_REF] Abdellaoui | Nonlinear fractional Laplacian problems with nonlocal "gradient terms[END_REF]. In Chapter 2 we pursue the study of [START_REF] Abdellaoui | Nonlinear fractional Laplacian problems with nonlocal "gradient terms[END_REF] in the case where c λ 0 a.e. in Ω. More precisely, we address the situation where the three coefficients functions c λ , µ and h may change sign. As observed in [START_REF] Jeanjean | Multiple solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF], the structure of the set of solutions to [START_REF] Abdellaoui | Nonlinear fractional Laplacian problems with nonlocal "gradient terms[END_REF] depends on the size of c + λ but it is not affected by the size of c - λ . Let us then write c λ under the form c λ = λc +c -where we recall λ is a real parameter. We consider the boundary value problem

-∆u = (λc + (x) -c -(x))u + µ(x)|∇u| 2 + h(x) , u ∈ H 1 0 (Ω) ∩ L ∞ (Ω) , ( 21
)
under the assumption

                     Ω ⊂ R N , N ≥ 2, is a bounded domain with boundary ∂Ω of class C 1,1 , c + , c -, h + belong to L q (Ω) for some q > N /2 and µ, h -belong to L ∞ (Ω) , c + (x) ≥ 0, c -(x) ≥ 0 and c -(x)c + (x) = 0 a.e. in Ω, |Ω + | > 0,
where

Ω + := Supp(c + ), there exists a ε > 0 such that µ(x) ≥ µ 1 > 0 and c -≡ 0 in Ω 1 := {x ∈ Ω : d(x, Ω + ) < ε}. ( 22
)
Remark 0. Figure 3: Illustration of assumption [START_REF]Global regularity for the free boundary in the obstacle problem for the fractional Laplacian[END_REF] As in [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF][START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF][START_REF] Souplet | A priori estimates and bifurcation of solutions for an elliptic equation with semidefinite critical growth in the gradient[END_REF] we obtain our existence results using a topological approach, relying thus on the derivation of an a priori bound. In that direction our main result is the following. Theorem 0.2.1. [Theorem 2.1.1, Chapter 2] Assume [START_REF]Global regularity for the free boundary in the obstacle problem for the fractional Laplacian[END_REF]. Then, for any Λ 2 > Λ 1 > 0, there exists a constant M > 0 such that, for each λ ∈ [Λ 1 , Λ 2 ], any solution to [START_REF] Barrios | Free boundary regularity in the parabolic fractional obstacle problem[END_REF]

satisfies sup Ω u ≤ M.
Having at hand this a priori bound and following the strategy of [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF], we show the existence of a continuum of solutions to [START_REF] Barrios | Free boundary regularity in the parabolic fractional obstacle problem[END_REF]. More precisely, defining

Σ := {(λ, u) ∈ R × C(Ω) : u solves (21)}, (23)
and considering

-∆u = -c -(x)u + µ(x)|∇u| 2 + h(x) , u ∈ H 1 0 (Ω) ∩ L ∞ (Ω) , ( 24
)
which corresponds to [START_REF] Barrios | Free boundary regularity in the parabolic fractional obstacle problem[END_REF] with λ = 0, we prove the following theorem.

Theorem 0.2.2. [Theorem 2.1.2, Chapter 2] Assume [START_REF]Global regularity for the free boundary in the obstacle problem for the fractional Laplacian[END_REF] and suppose that (24) has a solution u 0 with c + u 0 0. Then, there exists a continuum C ⊂ Σ such that the projection of C on the λ-axis is an unbounded interval (-∞, λ] for some λ ∈ (0, +∞) and C bifurcates from infinity to the right of the axis λ = 0. Moreover:

1) for all λ ≤ 0, the problem (21) has an unique solution u λ and this solution satisfies u 0u 0 ∞ ≤ u λ ≤ u 0 ; 2) there exists λ 0 ∈ (0, λ] such that, for all λ ∈ (0, λ 0 ), the problem [START_REF] Barrios | Free boundary regularity in the parabolic fractional obstacle problem[END_REF] has at least two solutions with u λ,i ≥ u 0 for i = 1, 2. c) In Theorem 0.1.1 we give sufficient conditions to ensure the existence of a solution to [START_REF]Quelques propriétés des opérateurs elliptiques quasi linéaires[END_REF].

Λ 2 Λ 1 M u + ∞ λ λ 0
Let us now give some ideas about the proofs. As already mentioned, our main existence result, Theorem 0.2.2, relies on the derivation of an a priori bound on the solutions to [START_REF] Barrios | Free boundary regularity in the parabolic fractional obstacle problem[END_REF]. Since we do not have global sign conditions, the approaches used in [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF][START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF][START_REF] Souplet | A priori estimates and bifurcation of solutions for an elliptic equation with semidefinite critical growth in the gradient[END_REF] to obtain the a priori bound do not apply and a new strategy is required. To this aim, we further develop some techniques first sketched in the unpublished work [START_REF]Uniform bounds via regularity estimates for elliptic PDE with critical growth in the gradient[END_REF]. These techniques, in the framework of viscosity solutions and fully nonlinear equations, now lies at the heart of the paper [START_REF] Nornberg | A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient[END_REF]. We also make use of some ideas from [START_REF] García-Meli Án | A priori bounds and existence of solutions for slightly superlinear elliptic problems[END_REF].

First of all, we show that it is sufficient to control the behaviour of the solutions to [START_REF] Barrios | Free boundary regularity in the parabolic fractional obstacle problem[END_REF] in Ω + . This can be proved under a weaker assumption than [START_REF]Global regularity for the free boundary in the obstacle problem for the fractional Laplacian[END_REF]. More precisely, under the assumption

                 Ω ⊂ R N , N ≥ 2, is a bounded domain with boundary ∂Ω of class C 0,1 , c + , c -and h belong to L q (Ω) for some q > N /2 , µ belong to L ∞ (Ω) , c + (x) ≥ 0, c -(x) ≥ 0 and c -(x)c + (x) = 0 a.e. in Ω, |Ω + | > 0, where Ω + := Supp(c + ). ( 25
)
Lemma 0.2.3. [Lemma 2.4.1, Chapter 2] Assume that (25) holds. Then, there exists M > 0 such that, for any λ ∈ R, any solution u to [START_REF] Barrios | Free boundary regularity in the parabolic fractional obstacle problem[END_REF] satisfies

-sup Ω + u --M ≤ u ≤ sup Ω + u + + M.
Once we have this extra information, by compactness, we are lead to study what happens around a (unknown) point x ∈ Ω + . We shall consider separately the cases x ∈ Ω + ∩ Ω and x ∈ Ω + ∩ ∂Ω. A local analysis is then made respectively in a ball or a semiball centered at x. If similar analysis had previously been performed in other contexts when x ∈ Ω we believe it is not the case when x ∈ ∂Ω. For x ∈ Ω + ∩ Ω we have the following result. Lemma 0.2.4. [Lemma 2.4.2, Chapter 2] Assume that [START_REF]Global regularity for the free boundary in the obstacle problem for the fractional Laplacian[END_REF] holds and that x ∈ Ω + ∩ Ω. For each

Λ 2 > Λ 1 > 0, there exist M I > 0 and R > 0 such that, for any λ ∈ [Λ 1 , Λ 2 ], any solution u to (21) satisfies sup B R (x) u ≤ M I .
The proof of this result is based on the use of a local maximum principle that we borrow from [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF][START_REF] Malý | Fine regularity of solutions of elliptic partial differential equations[END_REF] (see Lemma 2.2.1 for the precise statement) and the classical weak Harnack inequality (see Lemma 2.2.3).

For x ∈ Ω + ∩ ∂Ω, the classical local estimates do not apply. The key to our approach is the use of a new boundary weak Harnack inequality. Actually, a major part of Chapter 2 is devoted to establish this inequality. This is done in a more general context than the one we need to treat [START_REF] Barrios | Free boundary regularity in the parabolic fractional obstacle problem[END_REF]. In particular, it also cover the case of the p-Laplacian with a zero order term. More precisely, for 1 < p < ∞, we consider the boundary value problem

-∆ p u + a(x)|u| p-2 u = 0 , u ∈ W 1,p 0 (ω) . ( 26
)
and we prove the following boundary weak Harnack inequality.

∈ ∂ω. Then, there exist R > 0, ε = ε(p, R, a ∞ , ω) > 0 and C = C(p, R, ε, a ∞ , ω) > 0 such that, for all R ∈ (0, R] , inf B R (x 0 )∩ω u(x) d(x, ∂ω) ≥ C B R (x 0 )∩ω u(x) d(x, ∂ω) ε dx 1/ε .
We believe this boundary weak Harnack inequality is of independent interest and will proved to be useful in other settings. Its proof uses ideas introduced by B. Sirakov [START_REF]Boundary Harnack estimates and quantitative strong maximum principles for uniformly elliptic PDE[END_REF]. In [START_REF]Boundary Harnack estimates and quantitative strong maximum principles for uniformly elliptic PDE[END_REF] such type of inequalities is established for an uniformly elliptic operator and viscosity solutions. However, since our context is quite different, the result of [START_REF]Boundary Harnack estimates and quantitative strong maximum principles for uniformly elliptic PDE[END_REF] does not apply to our situation and we need to work out an adapted proof. In particular, we would like to mention Lemmas 2.3.6 and 2.3.7. The proofs we provide of these results present an alternative approach which is shorter and somehow simpler than the one developed in [START_REF]Boundary Harnack estimates and quantitative strong maximum principles for uniformly elliptic PDE[END_REF] to prove the corresponding results.

Having at hand Theorem 0.2.5, we obtain the counterpart of Lemma 0.2.4 for x ∈ Ω + ∩ ∂Ω. Namely, we prove the following result. Lemma 0.2.6. [Lemma 2.4.3, Chapter 2] Assume that [START_REF]Global regularity for the free boundary in the obstacle problem for the fractional Laplacian[END_REF] holds and that x ∈ Ω + ∩ ∂Ω. For each Λ 2 > Λ 1 > 0, there exist R > 0 and M B > 0 such that, for any λ ∈ [Λ 1 , Λ 2 ], any solution to [START_REF] Barrios | Free boundary regularity in the parabolic fractional obstacle problem[END_REF]

satisfies sup B R (x)∩Ω u ≤ M B .
The combination of Lemmas 0.2.3, 0.2.4 and 0.2.6 gives us the proof of Theorem 0.2.1. Once we have proved Theorem 0.2.1, in order to prove Theorem 0.2.2, we follow the approach first used in [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF]. Nevertheless, as we do not have global sign conditions (neither on the coefficients functions nor on the solutions), several difficulties appear. We refer to Chapter 2, Section 2.5 for more details.

We would like to end this section presenting a corollary of Theorem 0.2.5 that we hope will be useful in other settings. Consider the boundary value problem

-∆u + a(x)u = b(x) , u ∈ H 1 0 (ω) , ( 27
)
under the assumption

           ω ⊂ R N , N ≥ 2, is a bounded domain with boundary ∂ω of class C 1,1 , a ∈ L ∞ (ω) , b -∈ L p (ω) for some p > N and b + ∈ L 1 (ω) , a ≥ 0 a.e. in ω . ( 28
)
Corollary 0.2.7. [Corollary 2.3.9, Chapter 2] Under the assumption [START_REF] Brezis | Gagliardo-Nirenberg inequalities and non-inequalities: The full story[END_REF], assume that u ∈ H 1 (ω) is a non-negative upper solution to [START_REF] Brezis | Some simple nonlinear PDE's without solutions[END_REF] and let x 0 ∈ ∂ω. Then, there exist

R > 0, ε = ε(R, a ∞ , ω) > 0, C 1 = C 1 (R, ε, a ∞ , ω) > 0 and C 2 = C 2 (ω, a ∞ ) > 0 such that, for all R ∈ (0, R], inf B R (x 0 )∩ω u(x) d(x, ∂ω) ≥ C 1 B R (x 0 )∩ω u(x) d(x, ∂ω) ε dx 1/ε -C 2 b - L p (ω) .

Removing the "thick zero set" condition

In Chapter 3 we continue the study of (5) in the case where c λ 0 a.e. in Ω. At the expense of considering µ constant, we remove the "thick zero set" condition on c λ considered in Chapter 2. We write again c λ under the form c λ = λc +c -where λ is a real parameter and we consider the boundary value problem

-∆u = (λc + (x) -c -(x))u + µ|∇u| 2 + h(x) , u ∈ H 1 0 (Ω) ∩ L ∞ (Ω) , ( 29
)
under the assumption

           Ω ⊂ R N , N ≥ 2, is a bounded domain with ∂Ω of class C 0,1 ,
c + , c -, and h belong to L q (Ω) for some q > N /2, µ > 0, c + 0, c -≥ 0 and c + (x)c -(x) = 0 a.e. in Ω.

(

) 30
Remark 0.3.1. Since h does not have a sign, there is no loss of generality in assuming µ > 0.

Before we address the study of [START_REF] Brezis | H 1 versus C 1 local minimizers[END_REF] in the case where c λ = λc +c -may change sign, we completely characterize the limit coercive case c λ ≤ 0. More generally, let us consider

-∆u = -d(x)u + µ|∇u| 2 + h(x), u ∈ H 1 0 (Ω) ∩ L ∞ (Ω), (31)
under the assumption

           Ω ⊂ R N , N ≥ 2, is a bounded domain with ∂Ω of class C 0,1 ,
d and h belong to L q (Ω) for some q > N /2, µ > 0 and d ≥ 0, [START_REF] Caffarelli | Drift diffusion equations with fractional diffusion and the quasigeostrophic equation[END_REF] and define

m d :=          inf u∈W d Ω (|∇u| 2 -µh(x)u 2 )dx , if W d ∅ , + ∞ , if W d = ∅ , ( 33
)
where

W d := {w ∈ H 1 0 (Ω) : d(x)w(x) = 0 a.e.
in Ω, w H 1 0 (Ω) = 1}. We prove the following sharp result. Due to its importance in the rest of the chapter, let us make explicit an immediate corollary of the previous theorem. We consider the boundary value problem

-∆u = -c -(x)u + µ|∇u| 2 + h(x) , u ∈ H 1 0 (Ω) ∩ L ∞ (Ω) , (34)
which corresponds to [START_REF] Brezis | H 1 versus C 1 local minimizers[END_REF] with λ = 0, and we have the following result.

Corollary 0. Now, having at hand this information about the limit coercive case c λ ≤ 0, we turn to the study of the non-coercive case λ > 0. First, using mainly variational techniques, we prove the following theorem.

Theorem 0.3.3. [Theorem 3.1.3, Chapter 3] Assume [START_REF] Caffarelli | Interior regularity for fractional systems[END_REF] and suppose that (34) has a solution. Then, there exists Λ > 0 such that, for all 0 < λ < Λ, (29) has at least two solutions. Remark 0.3.3. This result improves and generalizes the main result obtained in [START_REF] Jeanjean | Multiple solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF].

Next, considering stronger regularity assumptions on the coefficient functions and combining lower and upper solution and variational techniques, we improve the conclusions of Theorem 0.3.3. We derive a more precise information on the structure of the set of solutions to (29) when λ > 0. Under the assumption

           Ω ⊂ R N , N ≥ 2, is a bounded domain with ∂Ω of class C 1,1 , c + , c -,
and h belong to L p (Ω) for some p > N , µ > 0, c + 0, c -≥ 0 and c + (x)c -(x) = 0 a.e. in Ω, [START_REF] Chen | Semilinear fractional elliptic equations involving measures[END_REF] we prove the following theorems.

Theorem 0.3.4. [Theorem 3.1.4, Chapter 3] Assume [START_REF] Chen | Semilinear fractional elliptic equations involving measures[END_REF] and suppose that (34) has a solution u 0 with c + u 0 0. Then, every u solution to [START_REF] Brezis | H 1 versus C 1 local minimizers[END_REF] with λ > 0 and c + u ≥ 0 satisfies u u 0 . Moreover, there exists λ ∈]0, +∞[, such that: [START_REF] Brezis | H 1 versus C 1 local minimizers[END_REF] has at least two solutions u λ,1 , u λ,2 ∈ C 1 0 (Ω) such that u λ,1 u 0 ;

• for every λ ∈]0, λ[ ,
• (29) with λ = λ has at least one solution u λ ∈ C 1 0 (Ω) such that u λ ≥ u 0 ;

• for λ > λ the problem [START_REF] Brezis | H 1 versus C 1 local minimizers[END_REF] has no solutions u such that c + u ≥ 0.

Theorem 0.3.5. [Theorem 3.1.5, Chapter 3] Assume [START_REF] Chen | Semilinear fractional elliptic equations involving measures[END_REF] and suppose that (34) has a solution u 0 with c + u 0 0. Then, for every λ > 0, the problem (29) has at least two solutions

u λ,1 , u λ,2 ∈ C 1 0 (Ω) such that u λ,1 u 0 . c + u 0 0 λ c + u 0 0 λ λ
µ > 0 constant instead of µ ∈ L ∞ (Ω) with µ(x) ≥ µ 1 > 0 in Ω,
Theorems 0.3.4 and 0.3.5 extend the main existence results of [START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF] to the case where c may change sign. Moreover, unlike [START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF], we do not assume global sign conditions on u 0 (solution to [START_REF]Variational methods and sub-and supersolutions[END_REF]). Hence, even in the case where c -≡ 0, i.e. c has a sign, our hypotheses are weaker than the corresponding ones in [START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF].

d) Theorem 0.3.4 removes the "thick zero set" condition on the support of c λ considered in Theorem 0.2.2 and gives somehow a more precise information. In turn, here µ is constant and we require a stronger regularity on c λ and h + .

Finally, we give sufficient conditions in terms of h ensuring that the hypotheses of Theorem 0.3.4 or of Theorem 0.3.5 are satisfied.

Corollary 0.3.6. [Corollary 3.1.6, Chapter 3] Assume [START_REF] Chen | Semilinear fractional elliptic equations involving measures[END_REF] and suppose that (34) has a solution:

• If h 0, then the conclusions of Theorem 0.3.4 hold.

• If h 0, then the conclusions of Theorem 0.3.5 hold. Remark 0.3.5. In the case where h 0, problem [START_REF]Variational methods and sub-and supersolutions[END_REF] has always a solution.

We provide now some ideas about the proofs of Theorems 0.3.3, 0.3.4 and 0.3.5. As in Chapter 1 we exploit here the fact that µ is a constant. Nevertheless, due to the indefinite sign of c λ = λc +c -, we have to overcome new difficulties.

First, using the fact that µ is constant, we perform a change of variable and reduce [START_REF] Brezis | H 1 versus C 1 local minimizers[END_REF] to an equivalent semilinear problem which presents a variational formulation. To that end, the key is the construction of a lower solution to [START_REF] Brezis | H 1 versus C 1 local minimizers[END_REF] below every upper solution to this problem. The fact that c λ has no sign causes several difficulties in this construction. See Chapter 3, Section 3.4 for more details.

Once we have this equivalent problem, we shall look for solutions as critical points of an associated functional I λ . When λ is positive the functional is unbounded from below and presents a concave-convex-type geometry. Furthermore, it does not satisfy an Ambrosetti-Rabinowitz-type condition and c λ and h have no sign. In this context, to prove that the Palais-Smale sequences are bounded may be challenging. Due to the indefinite sign of c λ , the approach introduced in Chapter 1 cannot be adapted. Our proof here is inspired by [START_REF] Jeanjean | Multiple solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF]. However, since we do not impose h 0, the proof is more involved. The role of the lower solution previously discussed is again crucial. We refer to Chapter 3, Section 3.5 for more details.

Having at hand the boundedness of the Palais-Smale sequences, we argue as in Chapter 1. Theorem 0.3.3 is proved using mainly variational techniques as in [START_REF] Jeanjean | Multiple solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF][START_REF] Jeanjean | Existence and multiplicity for elliptic problems with quadratic growth in the gradient[END_REF]. Nevertheless, since our hypotheses are weaker than the corresponding ones in [START_REF] Jeanjean | Multiple solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF][START_REF] Jeanjean | Existence and multiplicity for elliptic problems with quadratic growth in the gradient[END_REF], to prove that we have a mountainpass geometry becomes much more involved. In Theorems 0.3.4 and 0.3.5 we combine lower and upper solution arguments and variational techniques. In both theorems a first solution is obtained throughout the existence of well-ordered lower and upper solutions. This solution is further proved to be a local minimum. Then, we obtain a second solution by a mountain-pass type argument.

Another key ingredient in the proofs of Theorems 0.3.4 and 0.3.5 is the following estimate that can be seen as a combination of the Strong maximum principle and the Hopf's Lemma with unbounded lower order coefficients. Under the assumption

           Ω ⊂ R N , N ≥ 2, is a bounded domain with ∂Ω of class C 1,1 ,
c belongs to L p (Ω) and B = (B 1 , . . . , B N ) belongs to (L p (Ω)) N for some p > N , c ≥ 0, [START_REF]Semilinear fractional elliptic equations with gradient nonlinearity involving measures[END_REF] we obtain the following result.

Theorem 0.3.7. Assume [START_REF]Semilinear fractional elliptic equations with gradient nonlinearity involving measures[END_REF] and let u ∈ C 1 (Ω) be an upper solution to

-∆u + B(x), ∇u + c(x)u = 0, u ∈ H 1 0 (Ω). ( 37
)
Then, either u ≡ 0 or u 0.

Nonlocal "gradient terms"

In Chapter 4 we address a different topic. We turn to study existence and non-existence results for several nonlocal problems involving nonlocal "gradient terms".

In the last fifteen years, there has been an increasing interest in the study of partial differential equations involving integro-differential operators. In particular, the case of the fractional Laplacian has been widely studied and is nowadays a very active field of research. This is due not only to its mathematical richness but also to the fact that the fractional Laplacian has appeared in a great number of equations modeling real world phenomena, especially those which take into account nonlocal effects. Among others, let us mention applications in quasi-geostrophic flows [START_REF] Caffarelli | Drift diffusion equations with fractional diffusion and the quasigeostrophic equation[END_REF], quantum mechanic [START_REF] Laskin | Fractional quantum mechanics and lévy path integrals[END_REF], mathematical finances [START_REF] Applebaum | Lévy processes and stochastic calculus[END_REF][START_REF] Cont | Financial modelling with jump processes[END_REF], obstacle problems [START_REF] Barrios | Free boundary regularity in the parabolic fractional obstacle problem[END_REF][START_REF]Global regularity for the free boundary in the obstacle problem for the fractional Laplacian[END_REF][START_REF] Caffarelli | Regularity of solutions to the parabolic fractional obstacle problem[END_REF] and crystal dislocation [START_REF] Dipierro | Strongly nonlocal dislocation dynamics in crystals[END_REF][START_REF] Dipierro | Dislocation dynamics in crystals: a macroscopic theory in a fractional Laplace setting[END_REF][START_REF] Toland | The Peierls-Nabarro and Benjamin-Ono equations[END_REF].

The first aim of Chapter 4 is to discuss, depending on the real parameter λ > 0, the existence and non-existence of solutions to the Dirichlet problem

       (-∆) s u = µ(x) D 2 s (u) + λf (x) , in Ω, u = 0 , in R N \ Ω, (38)
under the assumption

           Ω ⊂ R N , N ≥ 2, is a bounded domain with ∂Ω of class C 2 , s ∈ (1/2, 1), f ∈ L m (Ω) for some m > N /2s and µ ∈ L ∞ (Ω). ( 39
)
Throughout the chapter, (-∆) s stands for the, by know classical, fractional Laplacian operator. For a smooth function u and s ∈ (0, 1), it can be defined as

(-∆) s u(x) := a N ,s p.v. R N u(x) -u(y) |x -y| N +2s dy, where a N ,s := R N 1 -cos(ξ 1 ) |ξ| N +2s dξ -1 = - 2 2s Γ N 2 + s π N 2 Γ (-s)
, is a normalization constant and "p.v." is an abbreviation for "in the principal value sense". In (38), D 2 s is a nonlocal diffusion term. It plays the role of the "gradient square" in the nonlocal case and is given by

D 2 s (u) = a N ,s 2 p.v. R N |u(x) -u(y)| 2 |x -y| N +2s dy . ( 40
)
Since they will not play a role in this thesis, we normalize the constants appearing in the definitions of (-∆) s and D 2 s and we omit the p.v. sense. However, let us stress that these constants guarantee that lim

s→1 -(-∆) s u(x) = -∆u(x), ∀ u ∈ C ∞ 0 (R N ), (41)
and

lim s→1 -D 2 s (u(x)) = |∇u(x)| 2 , ∀ u ∈ C ∞ 0 (R N ). (42)
We refer to [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF] and [START_REF] Caffarelli | Interior regularity for fractional systems[END_REF] respectively for a proof of ( 41) and [START_REF] Da Lio | n/p-harmonic maps: regularity for the sphere case[END_REF]. Hence, at least formally, if s → 1 - in [START_REF] Cuesta | Existence results for quasilinear problems via ordered sub-and supersolutions[END_REF], we recover the local problem

-∆u = µ(x)|∇u| 2 + λf (x), in Ω, u = 0, on ∂Ω. ( 43
)
Observe that this equation has played an essential role in Chapters 1, 2 and 3. It is precisely our model problem [START_REF] Hamid | On the connection between two quasilinear elliptic problems with source terms of order 0 or 1[END_REF] with c ≡ 0. In these chapters, the existence of solutions is proved using either a priori estimates or, when it is possible, a suitable change of variable to obtain an equivalent semilinear problem. However, neither of these techniques seem to be appropriate to deal with the nonlocal problem [START_REF] Cuesta | Existence results for quasilinear problems via ordered sub-and supersolutions[END_REF].

In the spirit of the existing results for the local case, see for instance Theorem 0.1.1, our first main result shows the existence of a weak solution to [START_REF] Cuesta | Existence results for quasilinear problems via ordered sub-and supersolutions[END_REF] under a smallness condition on λf . Theorem 0.4.1. [Theorem 4.1.1, Chapter 4] Assume that (39) holds. Then there exists λ * > 0 such that, for all 0 < λ ≤ λ * , [START_REF] Cuesta | Existence results for quasilinear problems via ordered sub-and supersolutions[END_REF]

has a weak solution u ∈ W s,2 0 (Ω) ∩ C 0,α (Ω) for some α > 0. Remark 0.4.1.
a) The definition of weak solutions to [START_REF] Cuesta | Existence results for quasilinear problems via ordered sub-and supersolutions[END_REF] is given in Definition 4.1.1 and the definition of W s,2 0 (Ω) will be introduced in Section 4.2.

b) For λf ≡ 0, u ≡ 0 is a solution to [START_REF] Cuesta | Existence results for quasilinear problems via ordered sub-and supersolutions[END_REF] that obviously belongs to W s,2 0 (Ω) ∩ C 0,α (Ω). Hence, there is no loss of generality to assume that λ > 0.

The counterpart of |∇u| 2 in (43) is played in (38) by D 2 s (u). This term appears in several applications. For instance, let us mention [START_REF] Caffarelli | Interior regularity for fractional systems[END_REF][START_REF] Millot | On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into spheres[END_REF][START_REF] Schikorra | Integro-differential harmonic maps into spheres[END_REF] where it naturally appears as the equivalent of |∇u| 2 when considering fractional harmonic maps into the sphere.

Let us now give some ideas of the proof of Theorem 0.4.1. The existence of solutions to [START_REF] Cuesta | Existence results for quasilinear problems via ordered sub-and supersolutions[END_REF] can be related to the regularity of the solutions to a linear equation of the form

       (-∆) s v = h(x), in Ω, v = 0, in R N \ Ω. (44)
In Chapter 4, Section 4.3, we obtain sharp Calder ón-Zygmund type regularity results for the fractional Poisson equation [START_REF] Damascelli | Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-Laplace equations[END_REF] with low integrability data. We believe these results are of independent interest and will be useful in other settings. Actually, Section 4.3 can be read as an independent part of Chapter 4. We refer the interested reader to Propositions 4.3.1, 4.3.3 and 4.3.4.

Having at hand suitable regularity results for [START_REF] Damascelli | Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-Laplace equations[END_REF] and inspired by [START_REF] Phuc | Morrey global bounds and quasilinear Riccati type equations below the natural exponent[END_REF]Section 6], we develop a fixed point argument to obtain a solution to [START_REF] Cuesta | Existence results for quasilinear problems via ordered sub-and supersolutions[END_REF]. Note that, due to the nonlocality of the operator and of the "gradient term", the approach of [START_REF] Phuc | Morrey global bounds and quasilinear Riccati type equations below the natural exponent[END_REF] has to be adapted significantly. In particular, the form of the set where we apply the fixed point argument seems to be new in the literature. We consider a subset of W s,1 0 (Ω) where, in some sense, we require more "differentiability" and more integrability. This extra "differentiability" is a purely nonlocal phenomenon and it is related with our regularity results for [START_REF] Damascelli | Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-Laplace equations[END_REF]. See Section 4.4 for more details.

Let us also stress that the restriction s ∈ (1/2, 1) comes from the regularity results of Section 4.3. If suitable regularity results for [START_REF] Damascelli | Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-Laplace equations[END_REF] with s ∈ (0, 1/2] were available, our fixed point argument would provide the same existence result Theorem 0.4.1 also for s ∈ (0, 1/2].

Next, we prove that the smallness condition imposed in Theorem 0.4.1 is somehow necessary. Theorem 0.4.2. [Theorem 4.1.2, Chapter 4] Assume [START_REF]Eigenvalue problems for the p-Laplacian with indefinite weights[END_REF] and suppose that µ(x) ≥ µ 1 > 0 and f + 0. Then there exists λ * * > 0 such that, for all λ > λ * * , [START_REF] Cuesta | Existence results for quasilinear problems via ordered sub-and supersolutions[END_REF] has no weak solutions in W s,2 0 (Ω). Remark 0.4.2. a) Observe that, if v is a solution to

       (-∆) s v = -µ(x) D 2 s (v) -λf (x) , in Ω, v = 0 , in R N \ Ω,
then u = -v is a solution to [START_REF] Cuesta | Existence results for quasilinear problems via ordered sub-and supersolutions[END_REF]. Hence, if µ(x) ≤ -µ 1 < 0 and f -0 we recover the same kind of non-existence result and the smallness condition is also required.

b) Since we do not use the regularity results of Section 4.3, the restriction s ∈ (1/2, 1) is not necessary in the proof of Theorem 0.4.2. The result holds for all s ∈ (0, 1).

Also, in order to show that the regularity imposed on the data f is almost optimal, we provide a counterexample to our existence result when the regularity condition on f is not satisfied. The proof makes use of the Hardy potential.

Theorem 0.4.3. [Theorem 4.1.3, Chapter 4] Let Ω ⊂ R N , N ≥ 2, be a bounded domain with ∂Ω of class C 2 , let s ∈ (0, 1) and let µ ∈ L ∞ (Ω) such that µ(x) ≥ µ 1 > 0. Then, for all 1 ≤ p < N 2s , there exists f ∈ L p (Ω) such that [START_REF] Cuesta | Existence results for quasilinear problems via ordered sub-and supersolutions[END_REF] has no weak solutions in W s,2 0 (Ω) for any λ > 0. Using the same kind of approach than in Theorem 0.4.1, i.e. regularity results for [START_REF] Damascelli | Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-Laplace equations[END_REF] and our fixed point argument, one can obtain existence results for related problems involving different nonlocal diffusion terms and different nonlinearities.

First, we deal with the Dirichlet problem

       (-∆) s u = µ(x) u D 2 s (u) + λf (x) , in Ω, u = 0 , in R N \ Ω. (45)
For µ(x) ≡ 1, this problem can be seen as a particular case of the fractional harmonic maps problem considered in [START_REF] Caffarelli | Interior regularity for fractional systems[END_REF][START_REF] Millot | On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into spheres[END_REF]. We derive the following existence result for λf small enough.

Theorem 0.4.4. [Theorem 4.1.4, Chapter 4] Assume that (39) holds. Then, there exists λ * > 0 such that, for all 0 < λ ≤ λ * , (45) has a weak solution u ∈ W s,2 0 (Ω) ∩ C 0,α (Ω) for some α > 0. Next, motivated by some other results on fractional harmonic maps into the sphere [START_REF] Da Lio | Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps[END_REF][START_REF] Da Lio | n/p-harmonic maps: regularity for the sphere case[END_REF] and some classical results of harmonic analysis [109, Chapter V], we consider a different diffusion term. Depending on the real parameter λ > 0, we study the existence of solutions to the Dirichlet problem

       (-∆) s u = µ(x)|(-∆) s 2 u| q + λf (x) , in Ω, u = 0 , in R N \ Ω, (46)
under the assumption Finally, for s ∈ (0, 1) and φ ∈ C ∞ 0 (R N ), following [START_REF] Ponce | Elliptic PDEs, measures and capacities. From the Poisson equations to nonlinear Thomas-Fermi problems[END_REF][START_REF] Shieh | On a new class of fractional partial differential equations[END_REF], we define the (distributional Riesz) fractional gradient of order s as the vector field ∇ s : R N → R given by

               Ω ⊂ R N , N ≥ 2, is a bounded domain with ∂Ω of class C 2 , f ∈ L m (Ω) for some m ≥ 1 and µ ∈ L ∞ (Ω), s ∈ (1/2, 1) and 1 < q < N (N -ms) + .
∇ s φ(x) := R N φ(x) -φ(y) |x -y| s x -y |x -y| dy |x -y| N , ∀ x ∈ R N . ( 48
)
Then we deal with the Dirichlet problem

       (-∆) s u = µ(x)|∇ s u| q + λf (x) , in Ω, u = 0 , in R N \ Ω. ( 49
)
Theorem 0.4.6. [Theorem 4.1.6, Chapter 4] Assume that (47) holds. Then there exists λ * > 0 such that, for all 0 < λ ≤ λ * , (49) has a weak solution u ∈ W s,1 0 (Ω).

1

Existence and multiplicity for elliptic p-Laplacian problems with critical growth in the gradient

Introduction and main results

Let ∆ p u = div(|∇u| p-2 ∇u) denote the p-Laplacian operator. We consider, for any 1 < p < ∞, the boundary value problem

-∆ p u = λc(x)|u| p-2 u + µ(x)|∇u| p + h(x) , u ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) , (P λ )
under the assumptions

           Ω ⊂ R N , N ≥ 2 , is a bounded domain with ∂Ω of class C 0,1 ,
c and h belong to L q (Ω) for some q > max{N /p, 1}, c 0 and µ ∈ L ∞ (Ω) .

(A 0 )
The study of quasilinear elliptic equations with a gradient dependence up to the critical growth |∇u| p was initiated by L. Boccardo, F. Murat and J.P. Puel in the 80's and it has been an active field of research until now. Under the condition λc(x) ≤ -α 0 < 0 for some α 0 > 0, which is now referred to as the coercive case, the existence of solutions is a particular case of the results of [START_REF] Boccardo | Existence de solutions faibles pour des équations elliptiques quasilinéaires à croissance quadratique[END_REF][START_REF]L ∞ estimate for some nonlinear elliptic partial differential equations and application to an existence result[END_REF][START_REF] Dall'aglio | Nonlinear elliptic equations with natural growth in general domains[END_REF]. The weakly coercive case (λ = 0) was studied in [START_REF] Ferone | Nonlinear problems having natural growth in the gradient: an existence result when the source terms are small[END_REF] where, for µh N /p small enough, the existence of an unique solution is obtained, see also [START_REF] Hamid | On the connection between two quasilinear elliptic problems with source terms of order 0 or 1[END_REF]. The limit coercive case, where one just require that λc(x) ≤ 0 and hence c may vanish only on some parts of Ω, is more complex and was left open until [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF]. In that paper, for the case p = 2, it was observed that, under the assumption (A 0 ), the existence of solutions to (P λ ) is not guaranteed. Sufficient conditions in order to ensure the existence of solution were given.

The case λc(x) 0 also remained unexplored until very recently. First, in [START_REF] Jeanjean | Existence and multiplicity for elliptic problems with quadratic growth in the gradient[END_REF] the authors studied problem (P λ ) with p = 2. Assuming λ > 0 and µh small enough, in an appropriate sense, they proved the existence of at least two solutions. This result has now be complemented in several ways. In [START_REF] Jeanjean | Multiple solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF] the existence of two solutions is obtained, allowing the function c to change sign with c + 0 but assuming h 0. In both [START_REF] Jeanjean | Multiple solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF][START_REF] Jeanjean | Existence and multiplicity for elliptic problems with quadratic growth in the gradient[END_REF] µ > 0 is assumed constant. In [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF] the restriction µ constant was removed but assuming that h 0. Finally, in [START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF], under stronger regularity on the coefficients, cases where µ is non constant and h is non-positive or has no sign were treated. Actually in [START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF], under different sets of assumptions, the authors clarify the structure of the set of solutions to (P λ ) in the non-coercive case. Now, concerning (P λ ) with p 2, the only results in the case λc 0 are, up to our knowledge, presented in [START_REF] Hamid | On the connection between two quasilinear elliptic problems with source terms of order 0 or 1[END_REF][START_REF] Iturriaga | A quasilinear problem without the Ambrosetti-Rabinowitz-type condition[END_REF]. In [START_REF] Iturriaga | A quasilinear problem without the Ambrosetti-Rabinowitz-type condition[END_REF] the case c constant and h ≡ 0 is covered and in [START_REF] Hamid | On the connection between two quasilinear elliptic problems with source terms of order 0 or 1[END_REF], the model equation is

-∆ p u = |∇u| p + λf (x)(1 + u) b , b ≥ p -1 and f 0.
To state our first main result let us define

m + p,λ :=          inf u∈W λ Ω |∇u| p - µ + ∞ p -1 p-1 h(x)|u| p dx, if W λ ∅ , + ∞, if W λ = ∅ ,
and

m - p,λ :=          inf u∈W λ Ω |∇u| p + µ - ∞ p -1 p-1 h(x)|u| p dx, if W λ ∅ , + ∞, if W λ = ∅ ,
where b) The assumption m + p,λ > 0 and m - p,λ > 0 connects the cases λc(x) ≤ -α 0 < 0 and λc(x) ≡ 0. In fact, in case λc(x) < 0 a.e. in Ω we have W λ = ∅ and hence m + p,λ = m - p,λ = +∞. On the other hand, if λc(x) ≡ 0, then W 0 = W 1,p 0 (Ω) and m + p,λ > 0 and m - p,λ > 0 holds for example under a smallness condition on µ + p-1 ∞ h + and µ -p-1 ∞ h -as in Appendix 1.10. In particular observe that m + p,λ > 0 and m - p,λ > 0 in case µ ≥ 0 and h ≤ 0 as well as in case µ ≤ 0 and h ≥ 0.

W λ := {w ∈ W
c) If h is either non-negative or non-positive our hypotheses correspond to the ones introduced in [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF] for p = 2. However, observe that for all u ∈ W 1,p 0 (Ω), it follows that

Ω |∇u| p - µ + ∞ p -1 p-1 h(x)|u| p dx ≥ Ω |∇u| p - µ + ∞ p -1 p-1 h + (x)|u| p dx
and

Ω |∇u| p + µ - ∞ p -1 p-1 h(x)|u| p dx ≥ Ω |∇u| p - µ - ∞ p -1 p-1 h -(x)|u| p dx.
Hence, if h does not have a sign, our hypotheses improve the ones introduced in [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF] even for p = 2.

In the rest of the chapter we assume that µ is constant. Namely, we replace (A 0 ) by

           Ω ⊂ R N , N ≥ 2 , is a bounded domain with ∂Ω of class C 0,1 ,
c and h belong to L q (Ω) for some q > max{N /p, 1}, c 0 and µ > 0 .

(A 1 )

Observe that there is no loss of generality in assuming µ > 0 since, if u is a solution to (P λ ) with µ < 0, then w = -u satisfies

-∆ p w = λc(x)|w| p-2 w -µ|∇w| p -h(x) .
In [START_REF] Arcoya | Remarks on the uniqueness for quasilinear elliptic equations with quadratic growth conditions[END_REF], for p = 2 but assuming only (A 0 ), the uniqueness of solutions when λ ≤ 0 was obtained as a direct consequence of a comparison principle, see [START_REF] Arcoya | Remarks on the uniqueness for quasilinear elliptic equations with quadratic growth conditions[END_REF]Corollary 3.1]. As we show in Remark 1.3.3, such kind of principle does not hold in general when p 2. Actually the issue of uniqueness for equations of the form of (P λ ) appears widely open. If partial results, assuming for example 1 < p ≤ 2 or λc(x) ≤ -α 0 < 0, seem reachable adapting existing techniques, see in particular [START_REF] Leonori | Comparison principles for p-Laplace equations with lower order terms[END_REF][START_REF] Porretta | On the comparison principle for p-Laplace type operators with first order terms[END_REF][START_REF] Pucci | The maximum principle[END_REF], a result covering the full generality of (P λ ) seems, so far, out of reach. Theorem 1.1.2 below, whose proof makes use of some ideas from [START_REF] Abdellaoui | Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential[END_REF], crucially relies on the assumption that µ is constant. It permits however to treat the limit case (P 0 ) which plays an important role in the chapter.

Theorem 1.1.2. Assume that (A 1 ) holds and suppose λ ≤ 0. Then (P λ ) has at most one solution.

Let us now introduce

m p := inf Ω |∇w| p - µ p -1 p-1 h(x)|w| p dx : w ∈ W 1,p 0 (Ω), w = 1 . (1.1.1)
We can state the following result.

Theorem 1.1.3. Assume that (A 1 ) holds. Then (P 0 ) has a solution if, and only if, m p > 0.

Theorem 1.1.3 provides, so to say, a characterization in term of a first eigenvalue of the existence of solution to (P 0 ). This result again improves, for µ constant, [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF] and it allows to observe that, in case h 0, (P 0 ) has always a solution while the case h 0 is the "worse" case for the existence of a solution. In case h changes sign, the negative part of h "helps" in order to have a solution to (P 0 ) as explained in Remark 1.1.1. We give in Appendix 1.10, sufficient conditions on h + in order to ensure m p > 0.

Remark 1.1.2. Observe that the sufficient part of Theorem 1.1.3 is direct. Indeed, if m p > 0 then m + p,0 > 0 and m - p,0 > 0 and Theorem 1.1.1 implies that (P 0 ) has a solution. Gluing together the previous results we obtain the following existence and uniqueness result for λ ≤ 0. Corollary 1.1.4. Assume that (A 1 ) holds and suppose that (P 0 ) has a solution. Then, for all λ ≤ 0, (P λ ) has an unique solution. Now, we turn to the study the non-coercive case, namely when λ > 0. First, using mainly variational techniques we prove the following result.

Theorem 1.1.5. Assume that (A 1 ) holds and suppose that (P 0 ) has a solution. Then there exists Λ > 0 such that, for any 0 < λ < Λ, (P λ ) has at least two solutions.

As we shall see in Corollary 1.9.4, the existence of a solution to (P 0 ) is, in some sense, necessary for the existence of a solution when λ > 0.

Next, considering stronger regularity assumptions, we derive informations on the structure of the set of solutions in the non-coercive case. These informations complement Theorem 1.1.5. We denote by γ 1 > 0 the first eigenvalue of the problem

-∆ p u = γc(x)|u| p-2 u , u ∈ W 1,p 0 (Ω) , (1.1.2)
and we introduce the following order notions.

Definition 1.1.1. For h 1 , h 2 ∈ L 1 (Ω) we write

• h 1 ≤ h 2 if h 1 (x) ≤ h 2 (x) for a.e. x ∈ Ω, • h 1 h 2 if h 1 ≤ h 2 and meas({x ∈ Ω : h 1 (x) < h 2 (x)}) > 0.
For u, v ∈ C 1 (Ω) we write

• u < v if, for all x ∈ Ω , u(x) < v(x),
• u v if u < v and, for all x ∈ ∂Ω, either u(x) < v(x), or, u(x) = v(x) and ∂u ∂ν (x) > ∂v ∂ν (x), where ν denotes the exterior unit normal.

Under the assumptions

           Ω ⊂ R N , N ≥ 2 , is a bounded domain with ∂Ω of class C 2 , c and h belong to L ∞ (Ω) , c 0 and µ > 0 , (A 2 )
we state the following theorem.

Theorem 1.1.6. Assume that (A 2 ) holds and suppose that (P 0 ) has a solution u 0 . Then:

• If h 0, for every λ > 0, (P λ ) has at least two solutions u 1 , u 2 with u 1 0.

• If h 0, then u 0 0 and there exists λ ∈ (0, γ 1 ) such that:

• for every 0 < λ < λ, (P λ ) has at least two solutions satisfying u i ≥ u 0 ;

• for λ = λ, (P λ ) has at least one solution satisfying u ≥ u 0 ; -•] for any λ > λ, (P λ ) has no non-negative solution. b) In the case µ < 0, we have the opposite result i.e., two solutions for every λ > 0 in case h 0 and, in case h 0, the existence of λ > 0 such that (P λ ) has at least two negative solutions, at least one negative solution or no non-positive solution according to 0 < λ < λ, λ = λ or λ > λ.

In case h 0, we know that for λ > λ, (P λ ) has no non-negative solution but this does not exclude the possibility of having negative or sign changing solutions. Actually, we are able to prove the following result changing a little the point of view. We consider the boundary value problem

-∆ p u = λc(x)|u| p-2 u + µ|∇u| p + kh(x) , u ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) , (P λ,k )
with a dependence in the size of h and we obtain the following result.

Theorem 1.1.7. Assume that (A 2 ) holds and that h 0. Let

k 0 = sup k ∈ [0, +∞) : ∀ w ∈ W 1,p 0 (Ω) , Ω |∇w| p - µ p -1 p-1 k h(x)|w| p dx > 0 .
Then:

• For all λ ∈ (0, γ 1 ), there exists k = k(λ) ∈ (0, k 0 ) such that, for all k ∈ (0, k), the problem (P λ,k ) has at least two solutions u 1 , u 2 with u i 0 and for all k > k, the problem (P λ,k ) has no solution. Moreover, the function k(λ) is non-increasing.

• For λ = γ 1 , the problem (P λ,k ) has a solution if and only if k = 0. In that case, the solution is unique and it is equal to 0.

• For all λ > γ 1 , there exist 0 < k1 ≤ k2 < +∞ such that, for all k ∈ (0, k1 ), the problem (P λ,k ) has at least two solutions with u λ,1 0 and min u λ,2 < 0, for all k > k2 , the problem (P λ,k ) has no solution and, in case k1 < k2 , for all k ∈ ( k1 , k2 ), the problem (P λ,k ) has at least one solution u with u 0 and min u < 0. Moreover, the function k 1 (λ) is non-decreasing. Let us now say some words about our proofs. First note that when µ is assumed constant it is possible to perform a Hopf-Cole change of variable. Introducing

v = p -1 µ e µ p-1 u -1 ,
we can check that u is a solution to (P λ ) if, and only if, v > -

p-1 µ is a solution to -∆ p v = λc(x)g(v) + 1 + µ p -1 v p-1 h(x) , v ∈ W 1,p 0 (Ω) , (1.1.3)
where g is an arbitrary function satisfying

g(s) = p -1 µ 1 + µ p -1 s ln 1 + µ p -1 s p-2 p -1 µ 1 + µ p -1 s ln 1 + µ p -1 s , if s > - p -1 µ .
Working with problem (1.1.3) presents the advantage that one may assume, with a suitable choice of g when s ≤ -p-1 µ , that it has a variational structure. Nevertheless, from this point we face several difficulties.

First, we need a control from below on the solutions to (1.1.3), i.e. having found a solution to (1.1.3) one needs to check that it satisfies v > -p-1 µ , in order to perform the opposite change of variable and obtain a solution to (P λ ). To that end, in Section 1.4, we prove the existence of a lower solution u λ to (P λ ) such that every upper solution β of (P λ ) satisfies β ≥ u λ . This allows us to transform the problem (1.1.3) in a new one, which has the advantage of being completely equivalent to (P λ ). Note that the existence of the lower solution ultimately relies on the existence of an a priori lower bound. See Lemma 1.4.1 for a more general result.

We denote by I λ the functional associated to the new problem, see (1.5.5) for a precise definition. The "geometry" of I λ crucially depends on the sign of λ. When λ ≤ 0 is it essentially coercive and one may search for a critical point as a global minimum. When λ > 0 the functional I λ becomes unbounded from below and presents something like a concave-convex geometry. Then, in trying to obtain a critical point, the fact that g is only slightly superlinear at infinity is a difficulty. It implies that I λ does not satisfies an Ambrosetti-Rabinowitz-type condition and proving that Palais-Smale or Cerami sequences are bounded may be challenging. In the case of the Laplacian, when p = 2, dealing with this issue is now relatively standard but for elliptic problems with a p-Laplacian things are more complex and we refer to [START_REF] De Figueiredo | Elliptic equations involving the p-Laplacian and a gradient term having natural growth[END_REF][START_REF] Iturriaga | A quasilinear problem without the Ambrosetti-Rabinowitz-type condition[END_REF][START_REF] Iturriaga | Quasilinear problems involving changing-sign nonlinearities without an Ambrosetti-Rabinowitz-type condition[END_REF][START_REF] Li | The existence of a nontrivial solution to a nonlinear elliptic boundary value problem of p-Laplacian type without the Ambrosetti-Rabinowitz condition[END_REF] in that direction. Note however that in these last works, it is always assumed a kind of homogeneity condition which is not available here. Consequently, some new ideas are required, see Section 1.8.

Having at hand the Cerami condition for I λ with λ > 0, in order to prove Theorems 1.1.5, 1.1.6 and 1.1.7, we shall look for critical points which are either local-minimum or of mountainpass type. In Theorem 1.1.5 the geometry of I λ is "simple" and permits to use only variational arguments. In Theorems 1.1.6 and 1.1.7 however it is not so clear, looking directly to I λ , where to search for critical points. We shall then make uses of lower and upper solutions arguments. In both theorems a first solution is obtained through the existence of well-ordered lower and upper solutions. This solution is further proved to be a local minimum of I λ and it is then possible to obtain a second solution by a mountain pass argument. Our approach here follows the strategy presented in [START_REF] Chang | A variant mountain pass lemma[END_REF][START_REF]Variational methods and sub-and supersolutions[END_REF][START_REF] De Figueiredo | A variational approach to superlinear elliptic problems[END_REF]. See also [START_REF] Ambrosetti | Combined effects of concave and convex nonlinearities in some elliptic problems[END_REF].

Finally, concerning Theorem 1.1.1, where µ is not assumed to be constant, we obtain our solution through the existence of lower and an upper solution which correspond to solutions to (P λ ) where µ =µ - ∞ and µ = µ + ∞ respectively, see Section 1.6.

The chapter is organized as follows. In Section 1.2, we recall preliminary general results that are used in the rest of the chapter. In Section 1.3, we give a comparison principle and prove the uniqueness result for λ ≤ 0. Section 1.4 is devoted to the existence of the lower solution. In Section 1.5, we construct the modified problem that we use to obtain the existence results. The coercive and limit-coercive cases, corresponding to λ ≤ 0 are studied in Section 1.6 where we prove Theorem 1.1.1. Theorem 1.1.3 which gives a necessary and sufficient condition to the existence of a solution to (P 0 ) is established in Section 1.7. In Section 1.8 we show that I λ has, for λ > 0 small, a mountain pass geometry and that the Cerami compactness condition holds. This permits to give the proof of Theorem 1.1.5. Section 1.9 contains the proofs of Theorems 1.1.6 and 1.1.7. Finally in an Appendix we give conditions on h + that ensure that m p > 0.

Notation.

* = +∞ in case p ≥ N . The norm in L ∞ (Ω) is u ∞ = esssup x∈Ω |u(x)|.
2) For v ∈ L 1 (Ω) we define v + = max(v, 0) and v -= max(-v, 0).

3) The space W 1,p 0 (Ω) is equipped with the norm u := Ω |∇u| p dx

1/p . 4) We denote R + = (0, +∞) and R -= (-∞, 0). 5) For a, b ∈ L 1 (Ω) we denote {a ≤ b} = {x ∈ Ω : a(x) ≤ b(x)} .

Preliminaries

In this section we present some definitions and known results which are going to play an important role throughout all the chapter. First of all, we present some results on lower and upper solutions adapted to our setting. Let us consider the problem

-∆ p u + H(x, u, ∇u) = f (x) , u ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) , (1.2.1)
where f belongs to L 1 (Ω) and

H : Ω × R × R N → R is a Carathéodory function. Definition 1.2.1. We say that α ∈ W 1,p (Ω) ∩ L ∞ (Ω) is a lower solution to (1.2.1) if α + ∈ W 1,p 0 (Ω) and, for all ϕ ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) with ϕ ≥ 0, if follows that Ω |∇α| p-2 ∇α∇ϕ dx + Ω H(x, α, ∇α)ϕ dx ≤ Ω f (x)ϕ dx . Similarly, β ∈ W 1,p (Ω) ∩ L ∞ (Ω) is an upper solution to (1.2.1) if β -∈ W 1,p 0 (Ω) and, for all ϕ ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) with ϕ ≥ 0, if follows that Ω |∇β| p-2 ∇β∇ϕ dx + Ω H(x, β, ∇β)ϕ dx ≥ Ω f (x)ϕ dx .
R + → R + and a function k ∈ L 1 (Ω) such that |H(x, s, ξ)| ≤ b(|s|)[k(x) + |ξ| p ], a.e. x ∈ Ω, ∀(s, ξ) ∈ R × R N .
If there exist a lower solution α and an upper solution β of (1.2.1) with α ≤ β, then there exists a solution u of (1.2.1) with α ≤ u ≤ β. Moreover, there exists u min (resp. u max ) minimum (resp. maximum) solution to (1.2.1) with α ≤ u min ≤ u max ≤ β and such that, every solution u of (1.2.1)

with α ≤ u ≤ β satisfies u min ≤ u ≤ u max .
Next, we state the strong comparison principle for the p-Laplacian.

Theorem 1.2.2. [88, Theorem 1.3] [40, Proposition 2.4] Assume that ∂Ω is of class C 2 and let f 1 , f 2 ∈ L ∞ (Ω) with f 2 f 1 ≥ 0. If u 1 , u 2 ∈ C 1,τ 0 (Ω) , 0 < τ ≤ 1 , are respectively solution to -∆ p u i = f i , in Ω , for i = 1, 2 , (P i )
such that u 2 = u 1 = 0 on ∂Ω. Then u 2 u 1 .

We need also the following anti-maximum principle.

Proposition 1.2.3. [67, Theorem 5.1] Let Ω ⊂ R N , N ≥ 2, a bounded domain with ∂Ω of class C 1,1 , c, h ∈ L ∞ (Ω)
, γ 1 the first eigenvalue of (1.1.2). If h 0, then there exists δ 0 > 0 such that, for all λ ∈ (γ 1 , γ 1 + δ 0 ), every solution w of

-∆ p w = λ c(x)|w| p-2 w + h(x), u ∈ W 1,p 0 (Ω) (1.2.2)
satisfies w 0.
The following result is the well known Picone's inequality for the p-Laplacian. We state it for completeness.

Proposition 1.2.4. [11, Theorem 1.1] Let u, v ∈ W 1,p (Ω) with u ≥ 0, v > 0 in Ω and u v ∈ L ∞ (Ω). Denote L(u, v) = |∇u| p + (p -1) u v p |∇v| p -p u v p-1 |∇v| p-2 ∇v∇u , R(u, v) = |∇u| p -∇ u p v p-1 |∇v| p-2 ∇v . Then, it follows that • L(u, v) = R(u, v) ≥ 0 a.e. in Ω.
• L(u, v) = 0 a.e. in Ω if, and only if, u = kv for some constant k ∈ R.

Now, we consider the boundary value problem

-∆ p v = g(x, v), v ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) , (1.2.3)
being g : Ω × R → R a Carathéodory function such that, for all s 0 > 0, there exists A > 0, with

|g(x, s)| ≤ A , a.e. x ∈ Ω , ∀ s ∈ [-s 0 , s 0 ] . (1.2.4)
This problem can be handled variationally. Let us consider the associated functional Φ :

W 1,p 0 (Ω) → R defined by Φ(v) := 1 p Ω |∇v| p dx - Ω G(x, v) dx , where G(x, s) := s 0 g(x, t) dt .
We can state the following result.

M := v ∈ W 1,p 0 (Ω) : α ≤ v ≤ β .
Then the infimum of Φ on M is achieved at some v, and such v is a solution to (1.2.3).

Definition 1.2.2. A lower solution α ∈ C 1 (Ω) is said to be strict if every solution u of (1.2.1) with u ≥ α satisfies u α.

Similarly, an upper solution β ∈ C 1 (Ω) is said to be strict if every solution u of (1.2.1) such that u ≤ β satisfies u β.

Corollary 1.2.6. Assume that (1.2.4) is valid and that α and β are strict lower and upper solutions to (1.2.3) belonging to C 1 (Ω) and satisfying α β. Then there exists a local minimizer v of the functional Φ in the C 1 0 -topology. Furthermore, this minimizer is a solution to (1.2.3) with α v β.

Proof. First of all observe that Proposition 1.2.5 implies the existence of v ∈ W

1,p 0 (Ω) solution to (1.2.3), which minimizes Φ on M := {v ∈ W 1,p 0 (Ω) : α ≤ v ≤ β}.
Moreover, since g is an L ∞ -Carathéodory function, the classical regularity results (see [START_REF] Dibenedetto | C 1+α local regularity of weak solutions of degenerate elliptic equations[END_REF][START_REF] Lieberman | Boundary regularity for solutions of degenerate elliptic equations[END_REF]) imply that v ∈ C 1,τ (Ω) for some 0 < τ < 1. Since the lower and the upper solutions are strict, it follows that α v β and so, there is a C 1 0 -neighbourhood of v in M. Hence, it follows that v minimizes locally Φ in the C 1 0 -topology.

Proposition 1.2.7. [52, Proposition 3.9] Assume that g satisfies the following growth condition

|g(x, s)| ≤ d (1 + |s| σ ), a.e. x ∈ Ω , all s ∈ R , for some σ ≤ p * -1 and some positive constant d. Let v ∈ W 1,p 0 (Ω) be a local minimizer of Φ for the C 1 0 -topology. Then v ∈ C 1,τ 0 (Ω) for some 0 < τ < 1 and v is a local minimizer of Φ in the W 1,p 0 -topology.
We now recall abstract results in order to find critical points of Φ other than local minima.

Definition 1.2.3. Let (X, • ) be a real Banach space with dual space (X * , • * ) and let Φ : X → R be a C 1 functional. The functional Φ satisfies the Cerami condition at level c ∈ R if, for any Cerami sequence at level c ∈ R, i.e. for any sequence {x n } ⊂ X with

Φ(x n ) → c and Φ (x n ) * (1 + x n ) → 0 ,
there exists a subsequence {x n k } strongly convergent in X.

Theorem 1.2.8. [61, Corollary 9, Section 1, Chapter IV] Let (X, • ) be a real Banach space. Suppose that Φ : X → R is a C 1 functional. Take two points e 1 , e 2 ∈ X and define

Γ := {ϕ ∈ C([0, 1], X) : ϕ(0) = e 1 , ϕ(1) = e 2 } ,
and c := inf ϕ∈Γ max t∈[0,1] Φ(ϕ(t)) .
Assume that Φ satisfies the Cerami condition at level c and that c > max{Φ(e 1 ), Φ(e 2 )} .

Then, there is a critical point of Φ at level c, i.e. there exists x 0 ∈ X such that Φ(x 0 ) = c and Φ (x 0 ) = 0.

Theorem 1.2.9. [65, Corollary 1.6] Let (X, • ) be a real Banach space and let Φ :

X → R be a C 1 functional. Suppose that u 0 ∈ X is a local minimum, i.e. there exists ε > 0 such that Φ(u 0 ) ≤ Φ(u), for u -u 0 ≤ ε ,
and assume that Φ satisfies the Cerami condition at any level d ∈ R. Then, the following alternative holds:

i) either there exists 0 < γ < ε such that inf{Φ(u) :

u -u 0 = γ} > Φ(u 0 ),
ii) or, for each 0 < γ < ε, Φ has a local minimum at a point u γ with u γ -u 0 = γ and Φ(u γ ) = Φ(u 0 ).

Remark 1.2.1. In [START_REF] Ghoussoub | Duality and perturbation methods in critical point theory[END_REF], Theorem 1.2.9 is proved assuming the Palais-Smale condition which is stronger than our Cerami condition. Nevertheless, modifying slightly the proof, it is possible to obtain the same result with the Cerami condition.

Comparison principle and uniqueness results

In this section, we state a comparison principle and, as a consequence, we obtain uniqueness result for (P λ ) with λ ≤ 0, proving Theorem 1.1.2. Consider the boundary value problem

-∆ p u = µ|∇u| p + f (x, u) , u ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) , (1.3.1)
under the assumption

           Ω ⊂ R N , N ≥ 2 , is a bounded domain with ∂Ω of class C 0,1 , f : Ω × R → R is a L 1 -Carathéodory function with f (x, s) ≤ f (x, t) for a.e. x ∈ Ω and all t ≤ s, µ > 0. (1.3.2) Remark 1.3.1. As above, the assumption µ > 0 is not a restriction. If u ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) is a solution to (1.3.1) with µ < 0 then w = -u ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) is a solution to -∆ p w = -µ|∇w| p -f (x, -w) , w ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) , with -f (x, -s) satisfying the assumption (1.3.2).
Under a stronger regularity on the solutions, we can prove a comparison principle for (1.3.1). The proof relies on the Picone's inequality (Proposition 1.2.4) and is inspired by some ideas of [START_REF] Abdellaoui | Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential[END_REF]. Proof. Suppose that u 1 , u 2 are respectively a lower and an upper solution to (1.3.1). For simplicity denote t = pµ p-1 and consider as test function

Theorem 1.3.1. Assume that (1.3.2) holds. If u 1 , u 2 ∈ W 1,p (Ω) ∩ C(Ω)
ϕ = e tu 1 -e tu 2 + ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) . First of all, observe that ∇ϕ = t ∇u 1 e tu 1 -∇u 2 e tu 2 χ {u 1 >u 2 } ,
with χ A the characteristic function of the set A. Hence, using assumptions (1.3.2), it follows that

{u 1 >u 2 } |∇u 1 | p-2 ∇u 1 -|∇u 2 | p-2 ∇u 2 t ∇u 1 e tu 1 -t ∇u 2 e tu 2 -µ |∇u 1 | p -|∇u 2 | p e tu 1 -e tu 2 dx ≤ {u 1 >u 2 } f (x, u 1 ) -f (x, u 2 ) e tu 1 -e tu 2 dx ≤ 0.
Observe that

{u 1 >u 2 } |∇u 1 | p-2 ∇u 1 -|∇u 2 | p-2 ∇u 2 t ∇u 1 e tu 1 -t ∇u 2 e tu 2 dx -µ {u 1 >u 2 } |∇u 1 | p -|∇u 2 | p e tu 1 -e tu 2 dx = {u 1 >u 2 } e tu 1 |∇u 1 | p (t -µ) + µ|∇u 2 | p -t|∇u 2 | p-2 ∇u 2 ∇u 1 dx + {u 1 >u 2 } e tu 2 |∇u 2 | p (t -µ) + µ|∇u 1 | p -t|∇u 1 | p-2 ∇u 1 ∇u 2 dx. (1.3.3)
Next, as ∇e tu i = t ∇u i e tu i , i = 1, 2, we have

|∇u i | p = |∇e tu i | p t p e tpu i i = 1, 2 .
Hence, using the above identities, and as µ t-µ = p -1 and t t-µ = p, it follows that,

e tu 1 |∇u 1 | p (t -µ) + µ|∇u 2 | p -t|∇u 2 | p-2 ∇u 2 ∇u 1 = t -µ t p e t(p-1)u 1 |∇e tu 1 | p + (p -1) e tu 1 e tu 2 p |∇e tu 2 | p -p e tu 1 e tu 2 p-1 |∇e tu 2 | p-2 ∇e tu 2 ∇e tu 1 , e tu 2 |∇u 2 | p (t -µ) + µ|∇u 1 | p -t|∇u 1 | p-2 ∇u 1 ∇u 2 = t -µ t p e t(p-
e tu 1 = ke tu 2 in {u 1 > u 2 }. (1.3.5)
Since u 1 and u 2 are continuous on Ω and satisfy u 1u 2 ≤ 0 on ∂Ω, we deduce that

u 1 = u 2 on ∂{u 1 > u 2 }. Hence, (1.3.5) applied to x ∈ ∂{u 1 > u 2 }, implies k = 1. This implies that u 1 = u 2 in {u 1 > u 2 }, which proves u 1 ≤ u 2 , as desired. Corollary 1.3.2. Assume that (A 1 ) holds and suppose λ ≤ 0. If u 1 , u 2 ∈ W 1,p (Ω)∩C(Ω)
are respectively a lower and an upper solution to

(P λ ), then u 1 ≤ u 2 .
Proof. Define the function f :

Ω × R → R given by f (x, s) = λc(x)|s| p-2 s + h(x) .
Since (A 1 ) holds and λ ≤ 0, f is a L 1 -Carathéodory function which satisfies (1.3.2). Consequently, the proposition follows from Theorem 1.3.1

The following result guarantees the regularity that we need to apply the previous comparison principle.

Lemma 1.3.3. Assume that (A 1 ) holds and suppose λ ≤ 0. Then, any solution to (P λ ) belongs to C 0,τ (Ω).

Proof. This follows directly from [START_REF] Ladyzhenskaya | tseva, Linear and quasilinear elliptic equations[END_REF]

-∆u = µ|∇u| p + f (x, u) , u ∈ H 1 0 (Ω) ∩ L ∞ (Ω) , with 1 ≤ p ≤ 2 is proved in [17, Corollary 3.1]
. The following counter-example (see [98, p.7]) shows that there is no hope to obtain a similar result when p > 2. For N = 2 and R > 0, consider the problem on the ball

-∆ 4 u = |∇u| 2 in B(0, R) , u = 0 on ∂B(0, R) .
We easily see that u 1 = 0 and u 2 = 1 8 (R 2 -|x| 2 ) are both solutions to the above problem belonging to

W 1,4 0 (B(0, R)) ∩ L ∞ (B(0, R)).

A priori lower bound and existence of a lower solution

As explained in the introduction, the aim of this section is to find a lower solution below every upper solution to problem (P λ ). First of all, we show that under a rather mild assumption (in particular no sign on c is required) the solutions to (P λ ) admit a lower bound. Precisely we consider problem (P λ ) assuming now

           Ω ⊂ R N , N ≥ 2 , is a bounded domain with ∂Ω of class C 0,1 .
c and h belong to L q (Ω) for some q > max{N /p, 1},

µ ∈ L ∞ (Ω) satisfies 0 < µ 1 ≤ µ(x) ≤ µ 2 . (1.4.1)
Adapting the proof of [50, Lemma 3.1], based in turn on ideas of [START_REF] Abdellaoui | Elliptic problems with a Hardy potential and critical growth in the gradient: non-resonance and blow-up results[END_REF], we obtain Lemma 1.4.1. Under the assumptions (1.4.1), for any λ ≥ 0, there exists a constant M λ > 0 with

M λ := M(N , p, q, |Ω|, λ, µ 1 , c + q , h - q ) > 0 such that, every u ∈ W 1,p (Ω) ∩ L ∞ (Ω) upper solution to (P λ ) satisfies min Ω u > -M λ .
Proof. Let us split the proof in two steps.

Step 1: There exists a positive constant

M 1 = M 1 (p, q, N , |Ω|, λ, µ 1 , c + q , h - q ) > 0 such that u -≤ M 1 .
First of all, observe that for every function u ∈ W 1,p (Ω), it follows that

∇ (u -) p+1 p = p + 1 p (u -) 1/p ∇u -, and so, |∇u -| p u -= p p + 1 p ∇(u -) p+1 p p .
(

Suppose that u ∈ W 1,p (Ω) ∩ L ∞ (Ω) is an upper solution to (P λ ) and let us consider ϕ = u -as a test function. Under the assumptions (1.4.1), it follows that

- Ω |∇u -| p dx ≥ -λ Ω c(x)|u -| p dx + Ω µ(x)|∇u -| p u -dx + Ω h(x)u -dx ≥ -λ Ω c + (x)|u -| p dx + µ 1 Ω |∇u -| p u -dx - Ω h -(x)u -dx. (1.4.3)
By (1.4.2) and (1.4.3), we have that

µ 1 p p + 1 p Ω ∇(u -) p+1 p p dx + Ω |∇u -| p dx ≤ λ Ω c + (x)|u -| p dx + Ω h -(x)u -dx . (1.4.4)
Firstly, we apply Young's inequality and, for every ε > 0, it follows that

Ω c + (x)|u -| p dx = Ω (c + (x)) 1/p |u -| 1/p (c + (x)) p-1 p |u -| (p+1)(p-1) p dx ≤ C(ε) Ω c + (x)u -dx + ε Ω c + (x) (u -) p+1 p p dx
Moreover, applying Hölder and Sobolev inequalities, observe that

Ω c + (x) (u -) p+1 p p dx ≤ c + q (u -) p+1 p p qp q-1 ≤ S c + q ∇(u -) p+1 p p p
with S the constant from the embedding from W 1,p 0 (Ω) into L qp q-1 (Ω). Hence, choosing ε small enough to ensure that ε S λ c + q ≤ µ 1 2 p p+1 p and substituting in (1.4.4), we apply again Hölder and Sobolev inequalities and we find a constant C = C(µ 1 , λ, c + q , p, q, |Ω|, N ) such that

µ 1 2 p p + 1 p ∇(u -) p+1 p p p + ∇u -p p ≤ h - q + C(ε) c + q u -q q-1 ≤ C( h - q + c + q ) ∇u - p .
This allows to conclude that

u -≤ C h - q + c + q 1 p-1 =: M 1 .
Step 2: Conclusion.

Since (1.4.1) holds, every u ∈ W 1,p (Ω) ∩ L ∞ (Ω) upper solution to (P λ ) satisfies -∆ p u ≥ λc(x)|u| p-2 u -h -(x) , in Ω . (1.4.5)
Moreover, observe that 0 is also an upper solution to (1.4.5). Hence, since the minimum of two upper solution is an upper solution (see [START_REF] Cuesta | Existence results for quasilinear problems via ordered sub-and supersolutions[END_REF]Corollary 3.3]), it follows that min(u, 0) is an upper solution to (1.4.5). Furthermore, observe that min(u, 0) is an upper solution to

-∆ p u ≥ λc + (x)|u| p-2 u -h -(x) , in Ω .
Hence, applying [98, Theorem 6.1.2], we have the existence of

M 2 = M 2 (N , p, λ, |Ω|, c + q ) > 0 and M 3 = M 3 (N , p, λ, |Ω|, c + q ) > 0 such that sup Ω u -≤ M 2 u - p + h - q ≤ M 3 u -+ h - q .
Finally, the result follows by Step 1.

Remark 1.4.1.

a) Observe that the lower bound does not depend on h + and c -. In particular, we have the same lower bound for all h ≥ 0 and all c ≤ 0.

b) Since c does not have a sign, there is no loss of generality in assuming λ ≥ 0. If we consider λ ≤ 0, we recover the same result with M λ depending on c - q instead of c + q .

Proposition 1.4.2. Under the assumptions (A 1 ), for any λ ∈ R, there exists u λ ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) lower solution to (P λ ) such that, for every β upper solution to (P λ ), we have u λ ≤ min{0, β}.

Proof. We need to distinguish in our proof the cases λ ≤ 0 and λ ≥ 0. First we assume that λ ≤ 0. By Lemma 1.4.1, we have a constant M > 0 such that every upper solution β of (P λ ) satisfies β ≥ -M. Let α be the solution to

-∆ p u = -h -(x), u ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω).
It is then easy to prove that u = α-M ∈ W 1,p (Ω)∩L ∞ (Ω) is a lower solution to (P λ ) with u ≤ -M. By the choice of M, this implies that u ≤ u for every upper solution u of (P λ ). Now, when λ ≥ 0 we first introduce the auxiliary problem

       -∆ p u = λc(x)|u| p-2 u + µ|∇u| p -h -(x) -1, in Ω , u = 0 , on ∂Ω . (1.4.6)
Thanks to the previous lemma, there exists M λ > 0 such that, for every

β 1 ∈ W 1,p (Ω)∩L ∞ (Ω) upper solution to (1.4.6), we have β 1 ≥ -M λ . Now, for k > M λ , we introduce the problem        -∆ p u = -λc(x)k p-1 -h -(x) -1 , in Ω , u = 0 , on ∂Ω , (1.4.7)
and denote by α λ its solution. Since -λc(x)k p-1h -(x) -1 < 0, the comparison principle (see for instance [93, Lemma A.0.7]) implies that α λ ≤ 0. Observe that, for every β 1 upper solution to (P λ ), we have that

-∆ p β 1 ≥ λc(x)|β 1 | p-2 β 1 + µ|∇β 1 | p -h -(x) -1 ≥ -λc(x)k p-1 -h -(x) -1 = -∆ p α λ .
Consequently, it follows that

       -∆ p β 1 ≥ -∆ p α λ , in Ω , β 1 ≥ α λ = 0 , on ∂Ω,
and, applying again the comparison principle, that β 1 ≥ α λ . Now, we introduce the problem

       -∆ p u = λc(x)| T k (u)| p-2 T k (u) + µ|∇u| p -h -(x) -1, in Ω, u = 0 on ∂Ω, (1.4.8)
where

T k (s) = -k , if s ≤ -k , s , if s > -k .
Observe that β 1 and 0 are upper solutions to (1.4.8). Recalling that the minimum of two upper solution is an upper solution (see [START_REF] Cuesta | Existence results for quasilinear problems via ordered sub-and supersolutions[END_REF]Corollary 3.3]), it follows that β = min{0, β 1 } is an upper solution to (1.4.8). As α λ is a lower solution to (1.4.8) with α λ ≤ β, applying Theorem 1.2.1, we conclude the existence of u λ minimum solution to (1.4.8) with

α λ ≤ u λ ≤ β = min{0, β 1 }.
As, for every upper solution β of (P λ ), β is an upper solution to (1.4.8), we have α λ ≤ β. Recalling that u λ is the minimum solution to (1.4.8) with α λ ≤ u λ ≤ 0, we deduce that u λ ≤ β.

It remains to prove that u λ is a lower solution to (P λ ). First, observe that u λ is an upper solution to (1.4.6). By construction, this implies that u λ ≥ -M λ > -k. Consequently, u λ is a solution to (1.4.6) and so, a lower solution to (P λ ).

The Functional setting

Let us introduce some auxiliary functions which are going to play an important role in the rest of the chapter. Define

g(s) =              p -1 µ 1 + µ p -1 s ln 1 + µ p -1 s p-2 p -1 µ 1 + µ p -1 s ln 1 + µ p -1 s , s > - p -1 µ , 0 , s ≤ - p -1 µ , (1.5.1) G(s) = s 0 g(t) dt and H(s) = 1 p g(s)s -G(s) .
In the following lemma we prove some properties of these functions.

Lemma 1.5.1.

i) The function g is continuous on R, satisfies g > 0 on R + and there exists

D > 0 with -D ≤ g ≤ 0 on R -. Moreover, G ≥ 0 on R.
ii) For any δ > 0, there exists c = c(δ, µ, p) > 0 such that, for any s >

p-1 µ , g(s) ≤ c s p-1+δ .
iii) lim s→+∞ g(s)/s p-1 = +∞ and lim s→+∞ G(s)/s p = +∞.

iv) There exists R > 0 such that the function

H satisfies H(s) ≤ s t p-1 H(t), for R ≤ s ≤ t. v) The function H is bounded on R -.
Proof. i) By definition, it is obvious that g is continuous, g > 0 on R + and g is bounded and g ≤ 0 on R -. This implies also that G ≥ 0 by integration.

ii) First of all, recall that for any ε > 0 there exists c = c(ε) > 0 such that ln(s) ≤ c(ε)s ε for all s ∈ (1, ∞). This implies that, for any δ > 0,

lim s→+∞ g(s) s p-1+δ = lim s→+∞ (p -1)(1 + µ p-1 s) µs p-1 ln(1 + µ p-1 s) p-1 s δ = 0 .
Hence, there exists R > p-1 µ such that, for all s > R, g(s) s p-1+δ ≤ 1 .

As the function g(s)

s p-1+δ is continuous on the compact set [

p-1 µ , R], we have a constant C > 0 with g(s) s p-1+δ ≤ C on [ p-1 µ , R] .
The result follows for C = max(C, 1). s p = +∞ . iv) First of all, integrating by parts, we observe that, for any s ≥ 0 ,

G(s) = p -1 µ p 1 p 1 + µ p -1 s p ln 1 + µ p -1 s p-1 - µ p s 0 1 + µ p -1 t p-1 ln 1 + µ p -1 t p-2 dt ,
and so, for any s ≥ 0, it follows that

H(s) = 1 p p -1 µ p µ s 0 1 + µ p -1 t p-1 ln 1 + µ p -1 t p-2 dt -1 + µ p -1 s p-1 ln 1 + µ p -1 s p-1 .
To prove iv), we show that the function ϕ(s) :=

H(s) s p-1 is non-decreasing on [R, +∞) for some R > 0. Observe that ϕ (s) = 1 s p [H (s)s -(p -1)H(s)] .
Hence, we just need to prove that H (s)s-(p-1)H(s) ≥ 0 for s ≥ R. After some simple computations, we see that it is enough to prove the existence of R > 0 such that, for all s ≥ R, κ(s) ≥ 0 where

κ(s) = 1 + µ p -1 s p-2 ln 1 + µ p -1 s p-2 µs p -1 2 + ln 1 + µ p -1 s -µ s 0 1 + µ p -1 t p-1 ln 1 + µ p -1 t p-2 dt .
Observe that

κ (s) = µ p -1 1 + µ p -1 s p-3 ln 1 + µ p -1 s p-3 (p -2) µs p -1 -ln 1 + µ p -1 s 2 + µs p -1 2 ln 1 + µ p -1 s .
Hence, we distinguish two cases: i) In case p ≥ 2, it is obvious that κ (s) > 0, for any s > 0. This implies that κ is increasing and so, that κ(s) > 0 for s > 0, since κ(0) = 0.

ii) If 1 < p < 2, as lim s→∞ κ (s) = +∞, there exists R 1 > 0 such that, for any s ≥ R 1 , we have κ (s) > 1 and hence, there exists R 2 ≥ R 1 such that κ(s) > 0, for any s ≥ R 2 .

In any case, we can conclude the existence of R ≥ 0 such that κ(s) > 0 for any s ≥ R. Consequently, there exists R > 0 such that ϕ (s) > 0, for s ≥ R, which means that ϕ is non-decreasing for s ≥ R and hence H satisfies

H(s) ≤ s t p-1 H(t), for R ≤ s ≤ t.
v) This follows directly from the definition of the functions g and G.

Next, we define the function

α λ = p -1 µ e µ p-1 u λ -1 ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) , (1.5.2)
where

u λ ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω)
is the lower solution to (P λ ) obtained in Proposition 1.4.2. Before going further, since u λ ≤ 0, observe that 0 ≥ α λ ≥ -p-1 µ + ε for some ε > 0. Now, for any λ ∈ R, let us consider the auxiliary problem

-∆ p v = f λ (x, v) , v ∈ W 1,p 0 (Ω) , (Q λ )
where

f λ (x, s) =              λc(x)g(s) + 1 + µ p -1 s p-1 h(x) , if s ≥ α λ (x) , λc(x)g(α λ (x)) + 1 + µ p -1 α λ (x) p-1 h(x) , if s ≤ α λ (x) , (1.5.3)
where g is defined by (1.5.1). In the following lemma, we prove some properties of the solutions to (Q λ ).

Lemma 1.5.2. Assume that (A 1 ) holds. Then, it follows that:

i) Every solution to (Q λ ) belongs to L ∞ (Ω). ii) Every solution v of (Q λ ) satisfies v ≥ α λ . iii) A function v ∈ W 1,p 0 (Ω) is a solution to (Q λ ) if,

and only if, the function

u = p -1 µ ln 1 + µ p -1 v ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω)
is a solution to (P λ ).

Proof. i) This follows directly from [82, Theorem IV-7.1]. ii) First of all, observe that α λ is a lower solution to (Q λ ). For a solution v ∈ W 1,p 0 (Ω) of (Q λ ), we have v ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) by the previous step and, for all ϕ ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) with ϕ ≥ 0, Ω |∇v| p-2 ∇v -|∇α λ | p-2 ∇α λ ∇ϕ dx ≥ Ω f λ (x, v) -f λ (x, α λ ) ϕ dx. Now, since there exist constants d 1 , d 2 > 0 such that for all ξ, η ∈ R N , |ξ| p-2 ξ -|η| p-2 η , ξ -η ≥ d 1 (|ξ| + |η|) p-2 |ξ -η| 2 , if 1 < p < 2 , d 2 |ξ -η| p , if p ≥ 2 , ( 1.5.4)
(see for instance [93, Lemma A.0.5]), we choose ϕ = (α λv) + and obtain that 0

≥ {α λ ≥v} |∇v| p-2 ∇v -|∇α λ | p-2 ∇α λ ∇(α λ -v) dx ≥ {α λ ≥v} f λ (x, v) -f λ (x, α λ ) (α λ -v) dx = 0 .
Consequently, using again (1.5.4), we deduce that α λ = v in {α λ ≥ v} and so, that v ≥ α λ .

iii

) Suppose that v ∈ W 1,p 0 (Ω) is a solution to (Q λ ). The first parts, i), ii) imply that v ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) is such that v ≥ α λ ≥ - p-1 µ + ε with ε > 0 and hence u ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω). Let us prove that u is a (weak) solution to (P λ ). Let φ be an arbitrary function belonging to C ∞ 0 (Ω) and define ϕ = φ/(1 + µ p-1 v) p-1 . It follows that ϕ ∈ W 1,p 0 (Ω). As e µu p-1 = 1 + µ p-1 v, we have the following identity Ω |∇v| p-2 ∇v∇ϕ dx = Ω e µu |∇u| p-2 ∇u ∇φ 1 + µ p-1 v p-1 - µ φ ∇v 1 + µ p-1 v p dx = Ω e µu 1 + µ p-1 v p-1 |∇u| p-2 ∇u ∇φ - µφ∇ p-1 µ (e µ p-1 u -1) 1 + µ p-1 v dx = Ω |∇u| p-2 ∇u ∇φ -µφ∇u dx = Ω |∇u| p-2 ∇u∇φ dx -µ Ω |∇u| p φ dx.
On the other hand, by definition of g, observe that

Ω λc(x)g(v)+ 1 + µ p -1 v p-1 h(x) ϕ dx = Ω λc(x) p -1 µ ln 1 + µ p -1 v p-2 p -1 µ ln 1 + µ p -1 v + h(x) φ dx = Ω λc(x)|u| p-2 u + h(x) φ dx.
As v is a solution to (Q λ ) we deduce from these two identities that

Ω |∇u| p-2 ∇u∇φ dx = Ω λc(x)|u| p-2 u + µ|∇u| p + h(x) φ dx,
and so, u is a solution to (P λ ), as desired.

On the same way, assume that u ∈ W

1,p 0 (Ω) ∩ L ∞ (Ω) is a solution to (P λ ). By Proposition 1.4.2 we know that u ≥ u λ . Hence, it follows that v = p-1 µ e µu p-1 -1 ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) and satisfies v ≥ α λ ≥ - p-1 µ + ε for some ε > 0. Arguing exactly as before, we deduce that v is a solution to (Q λ ). Remark 1.5.1. Arguing exactly as in the proof of Lemma 1.5.2, iii), we can show that v 1 ∈ W 1,p (Ω)∩ L ∞ (Ω) (respectively v 2 ∈ W 1,p (Ω) ∩ L ∞ (Ω)
) is a lower solution (respectively an upper solution) of (Q λ ) if, and only if, the function

u 1 = p -1 µ ln 1 + µ p -1 v 1 respectively u 2 = p -1 µ ln 1 + µ p -1 v 2
is a lower solution (respectively an upper solution) of (P λ ).

The interest of problem (Q λ ) comes from the fact that it has a variational formulation. We can obtain the solutions to (Q λ ) as critical points of the functional I λ : W 1,p 0 (Ω) → R defined as

I λ (v) = 1 p Ω |∇v| p dx - Ω F λ (x, v)dx , (1.5.5)
where we define

F λ (x, s) = λc(x)G(s) + p -1 µp 1 + µ p -1 s p h(x), if s ≥ α λ (x) , (1.5.6)
and

F λ (x, s) = λc(x)g(α λ (x)) + 1 + µ p -1 α λ (x) p-1 h(x) (s -α λ ) + λc(x)G(α λ (x)) + p -1 µp 1 + µ p -1 α λ (x) p h(x), if s ≤ α λ (x) .
(1.5.7)

Observe that under the assumptions (A 1 ), since g has subcritical growth (see Lemma 1.5.1), I ∈ C 1 (W 1,p 0 (Ω), R) (see for example [START_REF] Dinca | Variational and topological methods for Dirichlet problems with p-Laplacian[END_REF] page 356). Lemma 1.5.3. Assume that (A 1 ) holds and let λ ∈ R be arbitrary, Then, any bounded Cerami sequence for I λ admits a convergent subsequence.

Proof. Let {v n } ⊂ W 1,p 0 (Ω) be a bounded Cerami sequence for I λ at level d ∈ R. We are going to show that, up to a subsequence,

v n → v ∈ W 1,p 0 (Ω) for a v ∈ W 1,p 0 (Ω). Since {v n } is a bounded sequence in W 1,p 0 (Ω), up to a subsequence, we can assume that v n v in W 1,p 0 (Ω), v n → v in L r (Ω), for 1 ≤ r < p * , and v n → v a.e. in Ω. First of all, recall that I λ (v n ), v n -v → 0 with I λ (v n ), v n -v = Ω |∇v n | p-2 ∇v n ∇(v n -v) dx - {v n ≥α λ } λc(x)g(v n )(v n -v) dx - {v n ≥α λ } 1 + µ p -1 v n p-1 (v n -v) h(x) dx - {v n ≤α λ } f λ (x, α λ (x))(v n -v) dx. Let 0 < δ < ( p N -1 q )p * , r < p * and s < p *
p-1+δ such that 1 q + 1 r + 1 s = 1. Using Lemma 1.5.1 ii), and the Sobolev embedding as well as Hölder inequality, we have that

λ {v n ≥α λ } c(x)g(v n )(v n -v) dx ≤ |λ| Ω |c(x)||g(v n )||v n -v| dx ≤ |λ| c q g(v n ) s v n -v r ≤ D|λ| c q 1 + v n p-1+δ (p-1+δ)s v n -v r ≤ DS|λ| c q 1 + v n p-1+δ v n -v r .
Since v n is bounded and v n → v in L r (Ω), for 1 ≤ r < p * , we obtain

λ {v n ≥α λ } c(x)g(v n )(v n -v) dx → 0 .
Arguing in the same way, we have

{v n ≥α λ } 1 + µ p -1 v n p-1 (v n -v) h(x) dx + {v n ≤α λ } f λ (x, α λ (x))(v n -v) dx → 0 .
So, we deduce that

Ω |∇v n | p-2 ∇v n ∇(v n -v) dx → 0 . (1.5.8)
Hence, applying [57, Theorem 10], we conclude that v n → v in W 1,p 0 (Ω), as desired.

Sharp existence results on the limit coercive case

In this section, following ideas from [18, Section 3], we prove Theorem 1.1.1. As a preliminary step, considering µ > 0 constant, we introduce

m p,λ :=          inf u∈W λ Ω |∇u| p - µ p -1 p-1 h(x)|u| p dx , if W λ ∅ , + ∞ , if W λ = ∅ .
where

W λ := {w ∈ W
I λ (u) ∈ R ∪ {-∞}
I λ (v n ) = +∞ .
Assume by contradiction that, along a subsequence, I λ (v n ) is bounded from above and hence lim sup

n→∞ I λ (v n ) v n p ≤ 0 . (1.6.1)
We introduce the sequence

w n = v n v n
, for all n ∈ N and observe that, up to a subsequence w n w weakly in W 1,p 0 (Ω), w n → w in L r (Ω), for 1 ≤ r < p * , and w n → w a.e. in Ω. We consider two cases: Case 1): w + W λ . In that case, the set Ω 0 = {x ∈ Ω : λc(x)w + (x) 0} ⊂ Ω has non-zero measure and so, it follows that v n (x) = w n (x) v n → ∞ a.e. in Ω 0 . Hence, taking into account that G ≥ 0 and lim s→+∞ G(s)/s p = +∞ (see Lemma 1.5.1) and using Fatou's Lemma, we have

lim sup n→∞ Ω λc(x)G(v n ) |v n | p |w n | p dx ≤ lim sup n→∞ Ω 0 λc(x)G(v n ) |v n | p |w n | p dx ≤ Ω 0 lim sup n→∞ λc(x)G(v n ) |v n | p |w n | p dx = -∞ . (1.6.2)
On the other hand, observe that for any v ∈ W 1,p 0 (Ω), we can rewrite

I λ (v) = 1 p Ω |∇v| p dx - Ω λc(x)G(v) dx + {v≤α λ } λc(x)G(v) dx - p -1 pµ {v≥α λ } 1 + µ p -1 v p h(x) dx - {v≤α λ } F λ (x, v) dx .
Hence, considering together (1.6.1) and (1.6.2), we obtain

0 ≥ lim sup n→∞ I λ (v n ) v n p ≥ lim inf n→∞ I λ (v n ) v n p ≥ -C -lim sup n→∞ Ω λc(x)G(v n ) v n p dx = +∞ ,
and so, Case 1) cannot occur.

Case 2): w + ∈ W λ . First of all, since λc ≤ 0 and G ≥ 0 (see Lemma 1.5.1), observe that for any v ∈ W 1,p 0 (Ω),

I λ (v) ≥ 1 p Ω |∇v| p - µ p -1 p-1 h(x)(v + ) p dx - 1 p µ p -1 p-1 {v≥α λ } h(x)(v -) p dx - p -1 µp {v≥α λ } 1 + µ p -1 v p - µ p -1 p |v| p h(x) dx - {v≤α λ } F λ (x, v) dx . Moreover, observe that 1 p {v≥α λ } 1 + µ p -1 v p - µ p -1 p |v| p h(x) dx = {v≥α λ } 1 0 s + µ p -1 v p-2 s + µ p -1 v ds h(x) dx ≤ Ω 1 + µ p -1 |v| p-1 |h(x)| dx ≤ D h q 1 + v p-1 , (1.6.3)
for some constant D > 0. Thus, for any v ∈ W 1,p 0 (Ω), it follows that

I λ (v) ≥ 1 p Ω |∇v| p - µ p -1 p-1 h(x)(v + ) p dx - 1 p µ p -1 p-1 {v≥α λ } h(x)(v -) p dx -D h q 1 + v p-1 - {v≤α λ } F λ (x, v) dx .
(1.6.4) Hence, using that by the definition of F λ (see (1.5.6) and (1.5.7)) there exists m ∈ L q (Ω) , q > max{N /p, 1}, such that, for a.e. x ∈ Ω and all s ≤ 0,

|F λ (x, s)| ≤ m(x)(1 + |s|) , (1.6.5)
and applying (1.6.1) and (1.6.4), we deduce, as

w + ∈ W λ , that 0 ≥ lim sup n→∞ I λ (v n ) v n p ≥ lim inf n→∞ I λ (v n ) v n p ≥ 1 p Ω |∇w| p - µ p -1 p-1 h(x)(w + ) p dx ≥ 1 p min{1, m p,λ } w p ≥ 0 ,
and so, that

lim n→∞ I λ (v n ) v n p = 0 and w ≡ 0 .
Finally, taking into account that w n → 0 in L r (Ω), for 1 ≤ r < p * , we obtain the contradiction

0 = lim n→∞ I λ (v n ) v n p ≥ 1 p .
Hence, Case 2) cannot occur.

Proof of Theorem 1.1.1. To prove this result, we look for a couple of lower and upper solutions (α, β) of (P λ ) with α ≤ β and then we apply Theorem 1.2.1. First, assume that both µ + ∞ > 0 and µ - ∞ > 0. Observe that any solution to

-∆ p u = λc(x)|u| p-2 u + µ + ∞ |∇u| p + h(x), u ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω), (1.6.6)
is an upper solution to (P λ ) and, any solution to

-∆ p u = λc(x)|u| p-2 u -µ - ∞ |∇u| p + h(x), u ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω), (1.6.7)
is a lower solution to (P λ ). Now, since m + p,λ > 0, Proposition 1.6.1 ensures the existence of β ∈ W 1,p 0 (Ω)∩L ∞ (Ω) solution to (1.6.6). In the same way, m - p,λ > 0 implies the existence of v ∈ W

1,p 0 (Ω)∩ L ∞ (Ω) solution to -∆ p v = λc(x)|v| p-2 v + µ - ∞ |∇v| p -h(x), v ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω),
and hence α = -v is a solution to (1.6.7). Moreover, Lemma 1.3.3 implies α, β ∈ W 1,p 0 (Ω) ∩ C(Ω). Hence, since α is a lower solution to (1.6.6), it follows that α ≤ β, thanks to Theorem 1.3.1. Thus, we can apply Theorem 1.2.1 to conclude the proof. Now note that if µ + ∞ = 0, (1.6.6) reduces to

-∆ p u = λc(x)|u| p-2 u + h(x), u ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω), (1.6.8)
which has a solution by [START_REF] Dinca | Variational and topological methods for Dirichlet problems with p-Laplacian[END_REF]Theorem 13]. This solution corresponds again to an upper solution to (P λ ). Similarly, we can justify the existence of the lower solution when µ - ∞ = 0.

A necessary and sufficient condition

In this section we prove Theorem 1.1.3. First of all, following the ideas of [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF], inspired in turn in ideas of [4], we find a necessary condition for the existence of a solution to (P 0 ). Recall that the problem (P 0 ) is given by

-∆ p u = µ|∇u| p + h(x) , u ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) . (P 0 )
Proposition 1.7.1. Assume that (A 1 ) holds and suppose that (P 0 ) has a solution. Then m p defined by (1.1.1) satisfies m p > 0.

Proof. Assume that (P 0 ) has a solution u ∈ W

1,p 0 (Ω) ∩ L ∞ (Ω). Then, for any φ ∈ C ∞ 0 (Ω), it follows that Ω |∇u| p-2 ∇u∇(|φ| p ) dx -µ Ω |∇u| p |φ| p dx - Ω h(x)|φ| p dx = 0 . (1.7.1)
Now, applying Young's inequality, observe that

Ω |∇u| p-2 ∇u∇(|φ| p ) dx = p Ω |φ| p-2 φ|∇u| p-2 ∇u∇φ dx ≤ p Ω |φ| p-1 |∇u| p-1 |∇φ| dx ≤ µ Ω |φ| p |∇u| p dx + p -1 µ p-1 Ω |∇φ| p dx .
Hence, substituting in (1.7.1), multiplying by

µ p-1 p-1
and using the density of

C ∞ 0 (Ω) in W 1,p 0 (Ω), we obtain Ω |∇φ| p - µ p -1 p-1 h(x)|φ| p dx ≥ 0 , ∀ φ ∈ W 1,p 0 (Ω) . (1.7.2)
Arguing by contradiction, assume that inf

       Ω |∇φ| p - µ p -1 p-1 h(x)|φ| p dx : φ ∈ W 1,p 0 (Ω), φ = 1        = 0.
By standard arguments there exists φ 0 ∈ C 0,τ (Ω) for some τ ∈ (0, 1), with φ 0 > 0 in Ω such that

Ω |∇φ 0 | p dx = µ p -1 p-1 Ω h(x)|φ 0 | p dx .
(1.7.3)

Now, substituting the above identity in (1.7.1) with φ = φ 0 , we have that

Ω |∇φ 0 | p + (p -1) µ p -1 p φ p 0 |∇u| p -p µ p -1 p-1 φ p-1 0 |∇u| p-2 ∇u∇φ 0 dx = 0 . ( 1

On the Cerami conditon and the Mountain-Pass Geometry

We are going to show that, for any λ > 0, the Cerami sequences for I λ at any level are bounded. The proof is inspired by [START_REF] Jeanjean | Existence and multiplicity for elliptic problems with quadratic growth in the gradient[END_REF], see also [START_REF] Jeanjean | On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazertype problem set on R N[END_REF]. Nevertheless it requires to develop some new ideas. In view of Lemma 1.5.3, this will imply that I λ satisfies the Cerami condition at any level d ∈ R.

Lemma 1.8.1. Fixed λ > 0 arbitrary, assume that (A 1 ) holds and suppose that m p > 0 with m p defined by (1.1.1). Then, the Cerami sequences for I λ at any level d ∈ R are bounded.

Proof. Let {v n } ⊂ W 1,p 0 (Ω) be a Cerami sequence for I λ at level d ∈ R. First we claim that {v - n } is bounded. Indeed since {v n } is a Cerami sequence, we have that

I λ (v n ), v - n = - Ω |∇v - n | p dx - Ω f λ (x, v n ) v - n dx → 0 (1.8.1)
from which, since f λ (x, s) is bounded on Ω × R -, the claim follows. To prove that {v + n } is also bounded we assume by contradiction that v n → ∞. We define

Ω + n = {x ∈ Ω : v n (x) ≥ 0} and Ω - n = Ω \ Ω + n ,
and introduce the sequence

{w n } ⊂ W 1,p 0 (Ω) given by w n = v n / v n . Observe that {w n } ⊂ W 1,p 0 (Ω) is bounded in W 1,p 0 (Ω).
Hence, up to a subsequence, it follows that w n w in W 1,p 0 (Ω), w n → w strongly in L r (Ω) for 1 ≤ r < p * , and w n → w a.e. in Ω. We split the proof in several steps.

Step 1: cw ≡ 0.

As v - n is bounded and by assumption v n → ∞, clearly w -≡ 0. It remains to show that cw + ≡ 0. Assume by contradiction that cw + 0 i.e., defining

Ω + := {x ∈ Ω : c(x)w(x) > 0}, we assume |Ω + | > 0. Since v n → ∞ and I λ (v n ), v n → 0, it follows that I λ (v n ), v n v n p → 0 . (1.8.2)
First of all, observe that

I λ (v n ), v n = v n p - µ p -1 p-1 Ω + n h(x)|v n | p dx - Ω + n λc(x)g(v n )v n dx - Ω - n f λ (x, v n ) v n dx - Ω + n 1 + µ p -1 v n p-1 - µ p -1 p-1 |v n | p-2 v n v n h(x) dx. (1.8.3) Now, since f λ (x, s) is bounded on Ω × R -, we deduce that 1 v n p Ω - n f λ (x, v n ) v n dx → 0 . (1.8.4)
Moreover, using that w n → w in L r (Ω), 1 ≤ r < p * , with w -≡ 0, we have

1 v n p Ω + n |v n | p h(x) dx = Ω + n |w n | p h(x) dx → Ω w p h(x) dx , (1.8.5)
Next, we are going to show that

1 v n p Ω + n 1 + µ p -1 v n p-1 - µ p -1 p-1 |v n | p-2 v n v n h(x) dx → 0 .
(1.8.6)

Observe that

Ω + n 1 + µ p -1 v n p-1 - µ p -1 p-1 v p-1 n v n h(x) dx = (p -1) Ω + n 1 0 s + µ p -1 v n p-2 ds v n h(x) dx .
We consider separately the case p ≥ 2 and the case 1 < p < 2. In case p ≥ 2, there exists D > 0 such that

Ω + n 1 0 s + µ p -1 v n p-2 ds v n h(x) dx ≤ p -1 µ Ω + n 1 + µ p -1 v n p-1 |h(x)| dx ≤ D h q (1 + v n p-1 ) .
On the other hand, in case 1 < p < 2, we have a constant D > 0 with

Ω + n 1 0 s + µ p -1 v n p-2 ds v n h(x) dx ≤ p -1 µ Ω + n µ p -1 v n p-1 |h(x)| dx ≤ D h q v n p-1 .
The claim (1.8.6) follows then directly from the above inequalities. So, substituting (1.8.3), (1.8.4), (1.8.5) and (1.8.6) in (1.8.2) and using that g is bounded on R -, we deduce that

λ Ω c(x) g(v n ) v p-1 n w p n dx → 1 - µ p -1 p-1 Ω w p h(x) dx.
(1.8.7)

Let us prove that this is a contradiction. By Lemma 1.5.1, we know that lim s→+∞ g(s)/s p-1 = +∞ and as w n → w > 0 a.e. in Ω + , it follows that

c(x) g(v n ) v p-1 n w p n → +∞ a.e. in Ω + .
Since |Ω + | > 0, we have

Ω + c(x) g(v n ) v p-1 n w p n dx → +∞ . (1.8.8)
On the other hand, as g ≥ 0 on R + ,

g(s)
s p-1 is bounded on R -and w n pq is bounded, we have

Ω\Ω + c(x) g(v n ) v p-1 n w p n dx ≥ -D.
(1.8.9)

So (1.8.8) and (1.8.9) together give a contradiction with (1.8.7). Consequently, we conclude that cw ≡ 0.

Step 2: Let us introduce a new functional J λ : W

1,p 0 (Ω) → R defined as J λ (v) = I λ (v) - p -1 µp {v≥α λ } 1 + µ p -1 v p - µ p -1 p |v| p h -(x) dx and let us introduce the sequence {z n } ⊂ W 1,p 0 (Ω) defined by z n = t n v n , where t n ∈ [0, 1] satisfies J λ (z n ) = max t∈[0,1] J λ (tv n ) ,
(if t n is not unique we choose its smallest possible value). We claim that

lim n→∞ J λ (z n ) = +∞ .
We argue again by contradiction. Suppose the existence of M < +∞ such that

lim inf n→∞ J λ (z n ) ≤ M, (1.8.10)
and introduce a sequence {k n } ⊂ W 1,p 0 (Ω), defined as

k n = 2pM m p 1 p w n = 2pM m p 1 p v n v n .
Let us prove, taking M bigger if necessary, that for n large enough we have

J λ (k n ) > 3 2 M . (1.8.11) As 2pM m p 1 p 1 v n ∈ [0, 1]
for n large enough, this will give the contradiction

3 2 M ≤ lim inf n→∞ J λ (k n ) ≤ lim inf n→∞ J λ (z n ) ≤ M. First of all, observe that k n k := 2pM m p 1 p w in W 1,p 0 (Ω), k n → k in L r (Ω)
, for 1 ≤ r < p * , and k n → k a.e. in Ω. By the properties of G (see Lemma 1.5.1) together with k ≥ 0 and c k ≡ 0, it is easy to prove that

{k n ≥α λ } c(x)G(k n ) dx → Ω c(x)G(k) dx = 0.
(1.8.12)

As w -≡ 0, we have χ - n → 0 a.e. in Ω where χ - n is the characteristic function of Ω - n . Recall (see (1.6.5)) that we have m ∈ L q (Ω) , q > max{N /p, 1}, such that, for a.e. x ∈ Ω and all s ≤ 0,

|F λ (x, s)| ≤ m(x)(1 + |s|) .

This implies that

{k n ≤α λ } F λ (x, k n ) dx → 0 ,
as well as

{k n ≤α λ } |k n | p h(x) dx → 0 . (1.8.13)
Taking into account (1.8.12) and (1.8.13) we obtain that

J λ (k n ) = 1 p Ω |∇k n | p - µ p -1 p-1 h(x)|k n | p dx - p -1 pµ {k n ≥α λ } 1 + µ p -1 k n p - µ p -1 p |k n | p h + (x) dx + o(1) . (1.8.14) Now, observe that, by definition of m p , 1 p Ω |∇k n | p - µ p -1 p-1 h(x)|k n | p dx ≥ 1 p m p k n p = 2M . (1.8.15)
Furthermore, arguing as in (1.6.3), observe that

1 p {k n ≥α λ } 1 + µ p -1 k n p - µ p -1 p |k n | p h + (x) dx ≤ C h + q 1 + 2pM m p p-1 p ,
where C is independent of M. This implies that

J λ (k n ) ≥ 2M -C h + q 1 + 2pM m p p-1 p + o(1) ,
and, taking M bigger if necessary, for any n ∈ N large enough, (1.8.11) follows.

Step 3: For n ∈ N large enough, t n ∈ (0, 1).

By the definition of J λ and using that

1 + µ p -1 s p - µ p -1 p s p ≥ 0 , ∀ s ≥ 0 , observe that J λ (v n ) ≤ I λ (v n ) - p -1 pµ {α λ ≤v n ≤0} 1 + µ p -1 v n p - µ p -1 p |v n | p h -(x) dx ≤ I λ (v n ) + 1 p µ p -1 p-1 {α λ ≤v n ≤0} |v n | p h -(x)dx ≤ I λ (v n ) + 1 p µ p -1 p-1 α λ p ∞ h - 1 .
Consequently, since I λ (v n ) → d, there exists D > 0 such that, for all n ∈ N, J λ (v n ) ≤ D. Thus, taking into account that J λ (0) = -p-1 pµ h + 1 and J λ (t n v n ) → +∞, we conclude that t n ∈ (0, 1) for n large enough.

Step 4: Conclusion.

First of all, as t n ∈ (0, 1) for n large enough, by the definition of z n , observe that J λ (z n ), z n = 0, for those n. Thus, it follows that

J λ (z n ) = J λ (z n ) - 1 p J λ (z n ), z n = λ {z n ≥α λ } c(x)H(z n ) dx - p -1 pµ {z n ≥α λ } 1 + µ p -1 z n p-1 h + (x) dx - {z n ≤α λ } F λ (x, z n ) - 1 p f λ (x, z n ) z n dx.
Using the definition of f λ (x, s) for s ≤ α λ (x) and the fact that z - n is bounded, we easily deduce the existence of D 1 > 0 such that, for all n large enough,

p -1 pµ Ω 1 + µ p -1 z n p-2 1 + µ p -1 z n h + (x) dx ≤ -J λ (z n ) + λ Ω c(x)H(z n ) dx + D 1 . (1.8.16)
Now, since {v n } is a Cerami sequence, observe that (again for n large enough)

d + 1 ≥ I λ (v n ) - 1 p I λ (v n ), v n = λ {v n ≥α λ } c(x)H(v n ) dx - p -1 pµ {v n ≥α λ } 1 + µ p -1 v n p-1 h(x) dx - {v n ≤α λ } F λ (x, v n ) - 1 p f λ (x, v n ) v n dx
and, as above, there exists a constant D 2 > 0 such that

λ Ω c(x)H(v n ) dx ≤ p -1 pµ Ω 1 + µ p -1 v n p-2 1 + µ p -1 v n h(x) dx + D 2 .
(1.8.17)

Moreover, observe that

Ω 1 + µ p -1 v n p-2 1 + µ p -1 v n h(x) dx = Ω 1 + µ p -1 v n p-2 1 + µ p -1 v n h + (x) dx - Ω 1 + µ p -1 v n p-2 1 + µ p -1 v n h -(x) dx = 1 t p-1 n Ω t n + µ p -1 z n p-2 t n + µ p -1 z n h + (x) dx - Ω 1 + µ p -1 v n p-2 1 + µ p -1 v n h -(x) dx ≤ 1 t p-1 n Ω 1 + µ p -1 z n p-2 1 + µ p -1 z n h + (x) dx - Ω 1 + µ p -1 v n p-2 1 + µ p -1 v n h -(x) dx .
Considering together this inequality with (1.8.16) and (1.8.17), we obtain that

λ Ω c(x)H(v n ) dx ≤D 2 - J λ (z n ) t p-1 n + λ t p-1 n Ω c(x)H(z n ) dx + D 1 t p-1 n - p -1 pµ Ω 1 + µ p -1 v n p-2 1 + µ p -1 v n h -(x) dx.
(1.8.18)

Now, since H is bounded on R -, there exists D 3 > 0 such that, for all n ∈ N,

Ω - n c(x)H(z n ) dx ≤ D 3 . (1.8.19)
On the other hand, using iv) of Lemma 1.5.1, it follows that

Ω + n c(x)H(z n ) dx ≤ t p-1 n Ω + n c(x)H(v n ) dx + D 4 , ( 1
λ Ω - n c(x)H(v n ) dx ≤ D 5 - J λ (z n ) -D 6 t p-1 n - p -1 pµ Ω 1 + µ p -1 v n p-2 1 + µ p -1 v n h -(x) dx.
Arguing as in the previous steps, observe that

Ω 1 + µ p -1 v n p-2 1 + µ p -1 v n h -(x) dx ≥ - Ω - n 1 + µ p -1 v n p-1 h -(x) dx ≥ -D 7 h - q 1 + v - n p-1 ,
and so, we have that

λ Ω - n c(x)H(v n ) dx ≤ D 5 - J λ (z n ) -D 6 t p-1 n + D 7 h - q 1 + v - n p-1 .
By

Step 1, we know that v - n is bounded and Step 4 shows that J λ (z n ) → ∞. Recall also that, by Step 4, t n ∈ (0, 1). This implies that λ

Ω - n c(x)H(v n ) dx → -∞ (1.8.21)
which contradicts the fact that H is bounded on R -. This allows to conclude that the Cerami sequences for I λ at level d ∈ R are bounded. Now, we turn to the verification of the mountain pass geometry when λ ≥ 0 is small. Lemma 1.8.2. Assume that (A 1 ) holds and suppose that m p > 0. For λ ≥ 0 small enough, there exists r > 0 such that

I λ (v) > I λ (0) for v = r.
Proof. For an arbitrary fixed r > 0, let v ∈ W 1,p 0 (Ω) be such that v = r. We can write

I λ (v) = 1 p Ω |∇v| p - µ p -1 p-1 (v + ) p h(x) dx - 1 p µ p -1 p-1 {α λ ≤v≤0} |v| p h(x) dx - p -1 pµ {v≥α λ } 1 + µ p -1 v p - µ p -1 p |v| p h(x) dx - {v≤α λ } h(x) 1 + µ p -1 α λ p-1 (v -α λ ) + p -1 pµ 1 + µ p -1 α λ p dx -λ {v≥α λ } c(x)G(v) dx + {v≤α λ } c(x) g(α λ )(v -α λ ) + G(α λ ) dx .
Now, observe that, as above,

{v≥α λ } 1 + µ p -1 v p - µ p -1 p |v| p h(x) dx ≤ p Ω 1 + µ p -1 |v| p-1 |h(x)| dx ≤ D 1 (1 + r p-1 ),
with D 1 independent of λ. In the same way, using the fact that α λ ∈ [-

p-1 µ , 0], we deduce that - 1 p µ p -1 p-1 {α λ ≤v≤0} |v| p h(x) dx ≤ D 2 , - {v≤α λ } h(x) 1 + µ p -1 α λ p-1 (v -α λ ) + p -1 pµ 1 + µ p -1 α λ p dx ≤ D 3 + D 4 r ,
with D 2 , D 3 and D 4 independent of λ. Finally, observe that

Ω |∇v| p - µ p -1 p-1 (v + ) p h(x) dx = Ω |∇v + | p - µ p -1 p-1 (v + ) p h(x) dx + Ω |∇v -| p dx ≥ m p v + p + v -p ≥ min{1, m p } v p = min{1, m p } r p .
So, we obtain that

I λ (v) ≥ 1 p min{1, m p } r p -D 1 r p-1 -D 4 r -D 5 , -λ {v≥α λ } c(x)G(v) dx + {v≤α λ } c(x) g(α λ )(v -α λ ) + G(α λ ) dx , (1.8.22)
where the constants D i are independent of λ. Moreover, observe that for r large enough,

1 p min{1, m p } r p -D 1 r p-1 -D 4 r -D 5 ≥ 1 2p min{1, m p } r p + I λ (0) . (1.8.23)
On the other hand, by Lemma 1.5.1, for every δ > 0,

{v≥α λ } c(x)G(v) dx + {v≤α λ } c(x) g(α λ )(v -α λ ) + G(α λ ) dx ≤ D 6 r p+δ + D 7 r + D 8 , (1.8.24)
for some constant D 6 , D 7 , D 8 independent of λ. Hence, for λ small enough, we have

λ {v≥α λ } c(x)G(v) dx + {v≤α λ } c(x) g(α λ )(v -α λ ) + G(α λ ) dx ≤ 1 4p min{1, m p } r p , ( 1
I λ (v) ≥ 1 4p min{1, m p } r p + I λ (0) > I λ (0) .
Lemma 1.8.3. Assume that (A 1 ) holds and that m p > 0. For any λ > 0, M > 0, and r > 0, there exists

w ∈ W 1,p 0 (Ω) such that w > r and I λ (w) ≤ -M.
Proof. Consider v ∈ C ∞ 0 (Ω) such that v ≥ 0 and cv 0 and let us take t ∈ R + , t ≥ 1. First of all, as α λ ≤ 0, observe that

I λ (tv) ≤ 1 p t p Ω |∇v| p - µ p -1 p-1 |v| p h(x) dx -λt p Ω c(x)v p G(tv) t p v p dx + p -1 pµ Ω 1 + µ p -1 tv p - µ p -1 p (tv) p h -(x) dx.
As above, we have

1 p Ω 1 + µ p -1 tv p - µ p -1 p (tv) p h -(x) dx ≤ t p-1 Ω 1 + µ p -1 v p-1 h -(x) dx.
Hence, we obtain

I λ (tv) ≤ t p       1 p Ω |∇v| p - µ p -1 p-1 |v| p h(x) dx -λ Ω c(x)v p G(tv) t p v p dx + 1 t p -1 µ 1 + µ p -1 v p-1 ∞ h - 1       .
Now, since by Lemma 1.5.1, we have

lim t→∞ λ Ω c(x)v p G(tv) (tv) p dx = ∞ ,
we deduce that lim t→∞ I λ (tv) = -∞ from which the lemma follows.

Proposition 1.8.4. Assume that (A 1 ) holds and suppose that m p > 0. Moreover, suppose that λ ≥ 0 is small enough in order to ensure that the conclusion of Lemma 1.8.2 holds. Then, I λ possesses a critical point v ∈ B(0, r) with I λ (v) ≤ I λ (0), which is a local minimum of I λ .

Proof. From Lemma 1.8.2, we see that there exists r > 0 such that

m := inf v∈B(0,r) I λ (v) ≤ I λ (0) and I λ (v) > I λ (0) if v = r . Let {v n } ⊂ B(0, r) be such that I λ (v n ) → m. Since {v n } is bounded, up to a subsequence, it follows that v n v ∈ W 1,p 0 (Ω)
. By the weak lower semicontinuity of the norm and of the functional

I λ , we have v ≤ lim inf n→∞ v n ≤ r and I λ (v) ≤ lim inf n→∞ I λ (v n ) = m ≤ I λ (0) . Finally, as I λ (v) > I λ (0) if v = r, we deduce that v ∈ B(0, r) is a local minimum of I λ .
Proof of Theorem 1.1.5. Assume that λ > 0 is small enough in order to ensure that the conclusion of Lemma 1.9 Proof of Theorems 1.1.6 and 1.1.7

In this section, we assume the stronger assumption (A 2 ). In that case, we are able to improve our results on the non-coercive case. Proposition 1.9.1. Assume that (A 2 ) holds with h 0. Then, for every λ > 0, there exists v ∈ C 1,τ 0 (Ω), for some 0 < τ < 1, with v 0, which is a local minimum of I λ in the W 1,p 0 -topology and a solution to (Q λ ) with v ≥ α λ (with α λ defined by (1.5.2)).

Proof. First of all, observe that, as h 0, we have m p > 0 and hence, by Theorem 1.1.3, (P 0 ) has a

solution u 0 ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω). By Lemma 1.5.2, v 0 = p -1 µ e µ p-1 u 0 -1 ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω),
is then a weak solution to

         -∆ p v 0 = 1 + µ p -1 v 0 p-1 h(x) 0, in Ω, v 0 = 0, on ∂Ω.
As moreover, 1+ [START_REF] Dibenedetto | C 1+α local regularity of weak solutions of degenerate elliptic equations[END_REF][START_REF] Lieberman | Boundary regularity for solutions of degenerate elliptic equations[END_REF] that v 0 ∈ C 1,τ 0 (Ω), for some τ ∈ (0, 1) and, by the strong maximum principle (see [START_REF] Ázquez | A strong maximum principle for some quasilinear elliptic equations[END_REF]), that v 0 0. Now, we split the rest of the proof in three steps.

µ p-1 v 0 p-1 h(x) ∈ L ∞ (Ω), it follows from
Step 1: 0 is a strict upper solution to (Q λ ).

Observe that 0 is an upper solution to (Q λ ). In order to prove that 0 is strict, let v ≤ 0 be a solution to (Q λ ). As g ≤ 0 on R -(see Lemma 1.5.1), it follows that v is a lower solution to (Q 0 ) and so, thanks to the comparison principle, see Corollary 1.3.2, v ≤ v 0 0. Hence, 0 is a strict upper solution to (Q λ ).

Step 2: (Q λ ) has a strict lower solution α 0.

By construction α = α λ -1 is a lower solution to (Q λ ). Moreover, as every solution v of (Q λ ) satisfies v ≥ α λ α, we conclude that α is a strict lower solution to (Q λ ). Proof of the first part of Theorem 1.1.6. By Proposition 1.9.1, there exists a first critical point, which is a local minimum of I λ . By Theorem 1.2.9 and since the Cerami condition holds, we have two options. If we are in the first case, then together with Lemma 1.8.3, we see that I λ has the mountain-pass geometry and by Theorem 1.2.8, we have the existence of a second solution. In the second case, we have directly the existence of a second solution to (Q λ ). Then by Lemma 1.5.2 we conclude to the existence of two solutions to (P λ ). Now, we consider the case h 0.

Lemma 1.9.2. Assume that (A 2 ) holds and suppose that h 0. Recall that γ 1 denotes the first eigenvalue of (1.1.2). It follows that:

i) For any 0 ≤ λ < γ 1 , any solution u of the problem (P λ ) satisfies u 0.

ii) For λ = γ 1 , the problem (P λ ) has no solution.

iii) For λ > γ 1 , the problem (P λ ) has no non-negative solution.

Proof. Observe first that, taking u -as test function in (P λ ), we obtain

- Ω |∇u -| p -λ c(x)|u -| p dx = Ω µ|∇u| p u -+ h(x)u -dx.
(1.9.1)

i) For λ < γ 1 , there exists ε > 0 such that, for every u ∈ W

1,p 0 (Ω), Ω |∇u| p -λc(x)|u| p dx ≥ ε u p .
Consequently, as h 0 and µ > 0, we have that

0 ≥ -ε u -p ≥ - Ω |∇u -| p -λc(x)|u -| p dx = Ω µ|∇u| p u -+ h(x)u -dx ≥ 0,
which implies that u -= 0 and so that u ≥ 0. Hence -∆ p u 0 and by the strong maximum principle (see [START_REF] Ázquez | A strong maximum principle for some quasilinear elliptic equations[END_REF]), we have u 0.

ii) In case λ = γ 1 we have, for every u ∈ W

1,p 0 (Ω), Ω |∇u| p -γ 1 c(x)|u| p dx ≥ 0. (1.9.2)
Assume by contradiction that (P λ ) has a solution u. By (1.9.1) and (1.9.2), and using that h 0 and µ > 0, we have in particular

Ω |∇u -| p -γ 1 c(x)|u -| p dx = 0.
This implies that u -= kϕ 1 for some k ∈ R and ϕ 1 the first eigenfunction of (1.1.2) and hence, either u ≡ 0 or u 0. As h 0, the first case cannot occur as 0 is not a solution to (P λ ). In the second case, as h 0, we have Ω h(x)u -dx > 0 which contradicts (1.9.1), (1.9.2) and µ > 0.

iii) Suppose by contradiction that u is a non-negative solution to (P λ ). As in the proof of i), we prove u 0 and hence, there exists

D 1 > 0 such that u ≥ D 1 d with d(x) = dist(x, ∂Ω). Let ϕ 1 > 0 be the first eigenfunction of (1.1.2). As ϕ 1 ∈ C 1 (Ω), we have D 2 > 0 such that ϕ 1 ≤ D 2 d. This implies that ϕ 1 u ∈ L ∞ (Ω) and ϕ p 1 u p-1 ∈ W 1,p 0 (Ω) with ∇ ϕ p 1 u p-1 = p ϕ 1 u p-1 ∇ϕ 1 -(p -1) ϕ 1 u p ∇u.
Hence we can take ϕ p 1 u p-1 as test function in (P λ ) and we have that

λ Ω c(x)ϕ p 1 dx + Ω µ|∇u| p + h(x) ϕ p 1 u p-1 dx = Ω ∇ ϕ p 1 u p-1 |∇u| p-2 ∇u dx .
On the other hand, applying Proposition 1.2.4, we obtain

γ 1 Ω c(x)ϕ p 1 dx = Ω |∇ϕ 1 | p dx ≥ Ω ∇ ϕ p 1 u p-1 |∇u| p-2 ∇u dx.
Consequently, gathering together both inequalities, we have the contradiction

0 ≥ (γ 1 -λ) Ω c(x)ϕ p 1 dx ≥ Ω [µ|∇u| p + h(x)] ϕ p 1 u p-1 dx > 0 .
(1.9.3)

Corollary 1.9.3. Assume that (A 2 ) holds. If, for some λ > 0, (P λ ) has a solution u λ ≥ 0 then (P 0 ) has a solution.

Proof. Observe that u λ is an upper solution to (P 0 ). By Proposition 1.4.2, we know that (P 0 ) has a lower solution α with α ≤ u λ . The conclusion follows from Theorem 1.2.1.

Corollary 1.9.4. Assume that (A 2 ) holds with h 0. If (P λ ) has a solution for some λ ∈ (0, γ 1 ), then (P 0 ) has a solution.

Proof. If (P λ ) has a solution u, by Lemma 1.9.2, we have u 0. The result follows from Corollary 1.9.3.

Proposition 1.9.5. Assume that (P 0 ) has a solution u 0 ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) and suppose that (A 2 ) holds with h 0. Then there exists λ < γ 1 such that: i) For every 0 < λ < λ, there exists v ∈ C 1,τ 0 (Ω), for some 0 < τ < 1, with v 0, which is a local minimum of I λ in the W 1,p 0 -topology and a solution to (Q λ ).

ii) For λ = λ, there exists u ∈ C 1,τ 0 (Ω), for some 0 < τ < 1, with u ≥ u 0 , which is a solution to (P λ ).

iii) For λ > λ, the problem (P λ ) has no non-negative solution.

Proof. Defining λ = sup{λ : (P λ ) has a non-negative solution u λ }, we directly obtain that, for λ > λ, the problem (P λ ) has no non-negative solution and, by Lemma 1.9.2 ii), we see that λ ≤ γ 1 . Moreover, arguing exactly as in the first part of Proposition 1.9.1, we deduce that

v 0 = p -1 µ e µ p-1 u 0 -1 ∈ W 1,p 0 (Ω) ∩ C 1 0 (Ω) satisfies v 0 0. Now, fix λ ∈ (0, λ).
Step 1: 0 is a strict lower solution to (Q λ ).

The proof of this step follows the corresponding one of Proposition 1.9.1.

Step 2: (Q λ ) has a strict upper solution.

By the definition of λ we can find δ ∈ (λ, λ) and a non-negative solution u δ of (P δ ). As above, we easily see that

v δ = p -1 µ e µ p-1 u δ -1 ∈ W 1,p 0 (Ω) ∩ C 1 0 (Ω)
is a non-negative upper solution to (Q λ ) and v δ 0. Moreover, if v is a solution to (Q λ ) with v ≤ v δ , Theorem 1.2.2 implies that v v δ . Hence, v δ is a strict upper solution to (P λ ).

Step 3: Proof of i).

The conclusion follows as in Proposition 1.9.1.

Step 4: Existence of a solution for λ = λ.

Let {λ n } be a sequence with λ n < λ and λ n → λ and {v n } be the corresponding sequence of minimum of I λ n obtained in i). This implies that I λ n (v n ), ϕ = 0 for all ϕ ∈ W 1,p 0 (Ω). By the above construction, we also have

I λ n (v n ) ≤ I λ n (0) = - p -1 pµ Ω h(x) dx.
Arguing exactly as in Lemmata 1.8.1 and 1.5.3, we prove easily the existence of v ∈ W

1,p 0 (Ω) such that v n → v in W 1,p 0 (Ω) with v a solution to (Q λ ) for λ = λ.
As v n ≥ 0 we obtain also v ≥ 0, and, by Lemma 1.5.2, we have the existence of a solution u of (P λ ) with u ≥ 0. As u is then an upper solution to (P 0 ), we conclude that u ≥ u 0 .

Step 5: λ < γ 1 .

As by Lemma 1.9.2, the problem (P λ ) has no solution for λ = γ 1 , this follows from Step 4.

Proof of the second part of Theorem 1.1.6. By Lemma 1.9.2, we have u 0 0. Let us consider λ ∈ (0, γ 1 ) given by Proposition 1.9.5. Hence, for λ < λ, there exists a first critical point u 1 , which is a local minimum of I λ . We then argue as in the proof of the first part to obtain the second solution u 2 of (P λ ). By Lemma 1.9.2, these two solutions satisfy u i 0 and, by Theorem 1.3.1, we conclude that u i ≥ u 0 . Now, for λ = λ, respectively λ > λ, the result follows respectively from Proposition 1.9.5 ii) and iii).

Proof of Theorem 1.1.7. Part 1: Case λ ∈ (0, γ 1 ).

Step 1: There exists k > 0 such that (P λ,k ) has at least one solution.

Let λ 0 ∈ (λ, γ 1 ) and δ small enough such that

λ 0 s p-1 ≥ λ p -1 µ 1 + µ p -1 s ln 1 + µ p -1 s p-1 , ∀ s ∈ [0, δ].
Define w as a solution to

-∆ p w = λ 0 c(x)|w| p-2 w + h(x), w ∈ W 1,p 0 (Ω).
As λ 0 < γ 1 , we have w 0.

For l small enough, β = lw satisfies 0 ≤ β ≤ δ and, for k such that

l p-1 ≥ 1 + µ p-1 δ p-1 k, it is easy to prove that β = p-1 µ ln 1 + µ p-1
β is an upper solution to (P λ,k ) with β ≥ 0. As 0 is a lower solution to (P λ,k ), the claim follows from Theorem 1.2.1.

Step 2: For k ≥ k 0 , the problem (P λ,k ) has no solution.

Let u be a solution to (P λ,k ). By Lemma 1.9.2, we have u 0. This implies that u is an upper solution to (P 0,k ). As 0 is a lower solution to (P 0,k ), by Theorem 1.2.1, the problem (P 0,k ) has a solution and hence, by Proposition 1.7.1, m p > 0 which means that k < k 0 . This implies that, for k ≥ k 0 , the problem (P λ,k ) has no solution.

Step 3: k = sup{k ∈ (0, k 0 ) : (P λ,k ) has at least one solution} < k 0 .

Assume by contradiction that k = k 0 . Let {k n } be an increasing sequence such that k n → k, k n ≥ 1 2 k and there exists {u n } a sequence of solutions to (P λ,k n ). As in the previous step we have that u n is an upper solution to (P 0, 1 2 k ). By Theorem 1.3.1, we know that u n ≥ u 0 with u 0 0 the solution to (P 0, 1 2 k ). Now, let φ ∈ W 1,p 0 (Ω) ∩ C 1 0 (Ω) with φ 0 and

p -1 µ p-1 Ω |∇φ| p dx = k 0 Ω h(x)φ p dx.
Using φ p as test function and applying Young inequality as in the proof of Proposition 1.7.1, it follows that

p -1 µ p-1 Ω |∇φ| p dx ≥ Ω |∇u n | p-2 ∇u n ∇(|φ| p ) dx -µ Ω |φ| p |∇u n | p dx = λ Ω c(x)|u n | p-2 u n φ p dx + k n Ω h(x)φ p dx ≥ λ Ω c(x)|u 0 | p-2 u 0 φ p dx + k n Ω h(x)φ p dx.
Passing to the limit, we have the contradiction

p -1 µ p-1 Ω |∇φ| p dx ≥ λ Ω c(x)|u 0 | p-2 u 0 φ p dx + p -1 µ p-1 Ω |∇φ| p dx.
Step 4: For k > k, the problem (P λ,k ) has no solution and for k < k, the problem (P λ,k ) has at least two solutions u 1 , u 2 with u i 0.

The first statement is obvious by definition of k. Now, for k < k, let k ∈ (k, k) such that (P λ, k ) has a solution ũ. By Lemma 1.9.2, we have ũ 0. Then, it is easy to observe that β 1 = k k 1 p-1 ũ and

β 2 = ũ are both upper solutions to (P λ,k ) with 0 β 1 β 2 .
Observe that 0 is a strict lower solution to (P λ,k ). As β 1 0 is an upper solution to (P λ,k ), by Theorem 1.2.1, the problem (P λ,k ) has a minimum solution u 1 with 0 u 1 ≤ β 1 .

In order to prove the existence of the second solution, observe that if β 2 is not strict, it means that (P λ,k ) has a solution u 2 with u 2 ≤ β 2 but u 2 β 2 . Then u 2 u 1 and we have our two solutions. If β 2 is strict, we argue as in the proof of Theorem 1.1.6.

Step 5: The function k(λ) is non-increasing.

Let us consider λ 1 < λ 2 , k < k(λ 2 ) and ũ 0 a solution to (P λ 2 , k ). It is easy to prove that ũ is an upper solution to (P λ 1 , k ). As 0 is a lower solution to (P λ 1 , k ) with 0 ≤ ũ, by Theorem 1.2.1, the problem (P λ 1 , k ) has a solution. This implies that k(λ 1 ) ≥ k(λ 2 ).

Part 2: Case λ = γ 1 .

By Lemma 1.9.2, we know that the problem (P γ 1 ) has no solution for k > 0. Moreover, by (1.9.1), we see that if (P γ 1 ) with h ≡ 0 has a non-trivial solution, then u 0 and hence, by the strong maximum principle u 0. Arguing as in the proof of iii) of Lemma 1.9.2, we obtain the same contradiction (1.9.3).

Part 3: Case λ > γ 1 .

Step 1: There exists k > 0 such that (P λ,k ) has at least one solution u 0. By Proposition 1.2.3 with h = h, there exists δ 0 > 0 such that, for λ ∈ (γ 1 , γ 1 + δ 0 ), the solution to -∆ p w = λ c(x)|w| p-2 w + h(x) , w ∈ W 1,p 0 (Ω) , (1.9.4) satisfies w 0. Let us fix λ 0 ∈ (γ 1 , min(γ 1 + δ 0 , λ)) and δ small enough such that

λ 0 |s| p-2 s ≥ λ p -1 µ 1 + µ p -1 s ln 1 + µ p -1 s p-2 p -1 µ 1 + µ p -1 s ln 1 + µ p -1 s , ∀ s ∈ [-δ, 0] .
Define w as a solution to

-∆ p w = λ 0 c(x)|w| p-2 w + h(x), u ∈ W 1,p 0 (Ω).
As γ 1 < λ 0 < γ 1 + δ 0 , we have w 0. For l small enough, β = lw satisfiesmin(δ,

p-1 µ ) < β ≤ 0 and, for k ≤ l p-1 , it is easy to prove that β = p-1 µ ln 1 + µ p-1
β is an upper solution to (P λ,k ) with β 0. By Proposition 1.4.2, (P λ,k ) has a lower solution α with α ≤ β and the claim follows from Theorem 1.2.1.

Step 2: For k large enough, the problem (P λ,k ) has no solution.

Otherwise, let u be a solution to (P λ,k ). By Lemma 1.4.1 and Remark 1.4.1, we have M λ > 0 such that, for all k > 0, the corresponding solution u satisfies u ≥ -M λ . Let φ ∈ C 1 0 (Ω) with φ 0. Using φ p as test function, by Young inequality as in the proof of Proposition 1.7.1, it follows that

p -1 µ p-1 Ω |∇φ| p dx ≥ Ω |∇u| p-2 ∇u∇(φ p ) dx -µ Ω φ p |∇u| p dx = λ Ω c(x)|u| p-2 uφ p dx + k Ω h(x)φ p dx ≥ -λM p-1 Ω c(x)φ p dx + k Ω h(x)φ p dx.
which is a contradiction for k large enough.

Step 3: Define k1 = sup{k > 0 : (P λ,k ) has at least one solution u 0}. For k < k1 , the problem (P λ,k ) has at least two solutions with u 1 0 and min u 2 < 0.

For k < k1 , let k ∈ (k, k1 ) such that (P λ, k ) has a solution ũ 0. It is then easy to observe that

β 1 = ũ and β 2 = k k 1 p-1 ũ
with α ≤ u 1 ≤ β 1 .
In order to prove the existence of the second solution, observe that if β 2 is not strict, it means that (P λ,k ) has a solution u 2 with u 2 ≤ β 2 but u 2 β 2 . Then u 2 u 1 and we have our two solutions. If β 2 is strict, we argue as in the proof of Theorem 1.1.6.

Step 4: Define k2 = sup{k > 0 : (P λ,k ) has at least one solution}. For k > k2 , the problem (P λ,k ) has no solution and, in case k1 < k2 , for all k ∈ ( k1 , k2 ), the problem (P λ,k ) has at least one solution u with u 0 and min u < 0.

The first statement follows directly from the definition of k2 . In case k1 < k2 , for k ∈ ( k1 , k2 ), let k ∈ (k, k2 ) such that (P λ, k ) has a solution ũ. Observe that ũ is an upper solution to (P λ,k ). Again, Proposition 1.4.2 gives us a lower solution α of (P λ,k ) with α ≤ ũ and hence, by Theorem 1.2.1, the problem (P λ,k ) has a solution u. By definition of k1 , we have that u 0 and by Lemma 1.9.2, we know that min u < 0.

Step 5: The function k1 (λ) is non-decreasing.

Let us consider λ 1 < λ 2 , k < k1 (λ 1 ) and u 0 a solution to (P λ 1 ,k ). It is easy to prove that u is an upper solution to (P λ 2 ,k ). Again, applying Proposition 1.4.2 and Theorem 1.2.1, we prove that the problem (P λ 2 ,k ) has a solution u 0. This implies that k1 (λ 1 ) ≤ k1 (λ 2 ).

1.10 Appendix. Sufficient conditions Lemma 1.10.1.

Given f ∈ L r (Ω), r > max{N /p, 1} if p N and 1 < r < ∞ if p = N , let us consider E f (u) = Ω |∇u| p -f (x)|u| p dx 1 p for an arbitrary u ∈ W 1,p 0 (Ω). It follows that: i) If 1 < p < N and f + N /p < S N , E f (u) is an equivalent norm in W 1,p 0 (Ω). ii) If p = N and f + r < S N ,r , E f (u) is an equivalent norm in W 1,p 0 (Ω). iii) If p > N and f + 1 < S N , E f (u) is an equivalent norm in W 1,p 0 (Ω).
where, for p N , S N denotes the optimal constant in the Sobolev inequality, i.e. Proof. We give the proof for 1 < p < N . The other cases can be done in the same way. First of all, by applying Hölder and Sobolev's inequalities, observe that, for any h ∈ L N p (Ω), it follows that

S N = inf ∇u

Ω h(x)|u| p dx ≤ h N p u p p * ≤ 1 S N h N p ∇u p p .
On the one hand, by using this inequality, observe that

Ω |∇u| p -f (x)|u| p dx ≤ u p 1 + f N p S N .
On the other hand, following the same argument, we obtain that

Ω |∇u| p -f (x)|u| p dx ≥ Ω |∇u| p -f + (x)|u| p dx ≥ u p 1 - f + N p S N = A u p with A > 0 since f + N p < S N . The result follows.
As an immediate Corollary, we have a sufficient condition to ensure that m p > 0.

Corollary 1.10.2. Recall that m p is defined by (1.1.1). Under the assumptions (A 1 ), it follows that:

i) If 1 < p < N , then h + N /p < p-1 µ p-1 S N implies m p > 0. ii) If p = N , then h + q < p-1 µ p-1 S N ,q implies m p > 0. iii) If p > N , then h + 1 < p-1 µ p-1
S N implies m p > 0.

2

A priori bounds and multiplicity of solutions for an indefinite elliptic problem with critical growth in the gradient

Introduction and main results

The chapter deals with the existence and multiplicity of solutions for boundary value problems of the form

-∆u = c λ (x)u + µ(x)|∇u| 2 + h(x) , u ∈ H 1 0 (Ω) ∩ L ∞ (Ω) , (Q λ )
with c λ depending on a real parameter λ.

Here Ω ⊂ R N , N ≥ 2, is a bounded domain with boundary ∂Ω of class C 1,1 , c λ and h belong to L q (Ω) for some q > N /2 and µ belongs to L ∞ (Ω).

This type of problem, which started to be studied by L. Boccardo, F. Murat and J.P. Puel in the 80's, has attracted a new attention these last years. Under the condition c λ ≤ -α 0 < 0 a.e. in Ω for some α 0 > 0, the existence of a solution to (Q λ ) is a particular case of the results of [START_REF] Boccardo | Existence de solutions faibles pour des équations elliptiques quasilinéaires à croissance quadratique[END_REF][START_REF]L ∞ estimate for some nonlinear elliptic partial differential equations and application to an existence result[END_REF] and its uniqueness follows from [START_REF] Barles | Remarks on the maximum principle for nonlinear elliptic PDEs with quadratic growth conditions[END_REF][START_REF] Barles | Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions[END_REF]. The case c λ ≡ 0 was studied in [4,[START_REF] Ferone | Nonlinear problems having natural growth in the gradient: an existence result when the source terms are small[END_REF] and the existence requires some smallness condition on µh N /2 . The situation where one only requires c λ ≤ 0 a.e. in Ω (i.e. allowing parts of the domain where c λ ≡ 0 and parts of it where c λ < 0 ) proved to be more complex to treat. In the recent papers [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF][START_REF] Coster | Existence and multiplicity for elliptic p-Laplacian problems with critical growth in the gradient[END_REF], the authors explicit sufficient conditions for the existence of solutions to (Q λ ). Moreover, in [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF], the uniqueness of solution is established (see also [START_REF] Arcoya | Remarks on the uniqueness for quasilinear elliptic equations with quadratic growth conditions[END_REF] in that direction). All these results were obtained without requiring any sign conditions on µ and h.

In case c λ = λc 0, as we shall discuss later, problem (Q λ ) behaves very differently and becomes much richer. Following [START_REF] Sirakov | Solvability of uniformly elliptic fully nonlinear PDE[END_REF], which considers a particular case, [START_REF] Jeanjean | Existence and multiplicity for elliptic problems with quadratic growth in the gradient[END_REF] studied (Q λ ) with µ(x) ≡ µ > 0 and λc 0 but without a sign condition on h. The authors proved the existence of at least two solutions when λ > 0 and (µh) + N /2 are small enough. The restriction µ constant was removed in [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF] and extended to µ(x) ≥ µ 1 > 0 a.e. in Ω, at the expense of adding the hypothesis h 0. Next, in [START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF], assuming stronger regularity on c and h, the authors removed the condition h 0. In this paper, it is also lightened that the structure of the set of solutions when λ > 0, crucially depends on the sign of the (unique) solution to (Q 0 ). Note that, in [START_REF] Coster | Existence and multiplicity for elliptic p-Laplacian problems with critical growth in the gradient[END_REF], the above results are extended to the p-Laplacian case. Also, in the frame of viscosity solutions and fully nonlinear equations, under corresponding assumptions, similar conclusions have been obtained very recently in [START_REF] Nornberg | A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient[END_REF].

We refer to [START_REF] Jeanjean | Existence and multiplicity for elliptic problems with quadratic growth in the gradient[END_REF] for an heuristic explanation on how the behavior of (Q λ ) is affected by the change of sign in front of the linear term. Actually, in the case where µ(x) ≡ µ is a constant, it is possible to transform problem (Q λ ) into a new one which admits a variational formulation. When c λ ≤ -α 0 < 0, the associated functional, defined on H 1 0 (Ω), is coercive. If c λ 0, the coerciveness may be lost and when c λ 0, in fact as soon as c + λ 0, the functional is unbounded from below. In [START_REF] Jeanjean | Existence and multiplicity for elliptic problems with quadratic growth in the gradient[END_REF] this variational formulation was directly used to obtain the solutions. In [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF][START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF] where µ is non constant, topological arguments, relying on the derivation of a priori bounds for certain classes of solutions, were used.

The only known results where c λ may change sign are [START_REF]Existence and multiplicity for an elliptic problem with critical growth in the gradient and signchanging coefficients[END_REF][START_REF] Jeanjean | Multiple solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF] (see also [START_REF] García-Meli Án | A priori bounds and existence of solutions for slightly superlinear elliptic problems[END_REF] for related problems). They both concern the case where µ is a positive constant. In [START_REF] Jeanjean | Multiple solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF], assuming h 0, µh and c + λ small in an appropriate sense, the existence of at least two non-negative solutions was proved. In [START_REF]Existence and multiplicity for an elliptic problem with critical growth in the gradient and signchanging coefficients[END_REF], the authors show that the loss of positivity of the coefficient of u does not affect the structure of the set of solutions to (Q λ ) observed in [START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF] when c λ = λc 0. Since µ is constant in [START_REF]Existence and multiplicity for an elliptic problem with critical growth in the gradient and signchanging coefficients[END_REF][START_REF] Jeanjean | Multiple solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF], it is possible to treat the problem variationally. The main issue, to derive the existence of solutions, is then to show the boundedness of the Palais-Smale sequences.

When c λ 0, all the above mentioned results require either µ to be constant or to be uniformly bounded from below by a positive constant (or similarly bounded from above by a negative constant). In [START_REF] Souplet | A priori estimates and bifurcation of solutions for an elliptic equation with semidefinite critical growth in the gradient[END_REF], assuming that the three coefficients functions are non-negative, a first attempt to remove these restrictions on µ is presented. Following the approach of [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF], the proofs of the existence results reduce to obtaining a priori bounds on the non negative solutions to (Q λ ). First it is observed in [START_REF] Souplet | A priori estimates and bifurcation of solutions for an elliptic equation with semidefinite critical growth in the gradient[END_REF] that a necessary condition is the existence of a ball B(x 0 , ρ) ⊂ Ω and ν > 0 such that µ ≥ ν and c ≥ ν on B(x 0 , ρ). When N = 2 this condition also proves to be sufficient. If N = 3 or 4 the condition µ ≥ µ 0 > 0 on a set ω ⊂ Ω such that supp(c) ⊂ ω permits to obtain the a priori bounds. Other sets of conditions are presented when N = 3 and N = 5. However, if the approach developed in [START_REF] Souplet | A priori estimates and bifurcation of solutions for an elliptic equation with semidefinite critical growth in the gradient[END_REF], which relies on interpolation and elliptic estimates in weighted Lebesgue spaces, works well in low dimension, the possibility to extend it to dimension N ≥ 6 is not apparent.

In this chapter we pursue the study of (Q λ ) and consider situations where the three coefficients functions c λ , µ and h may change sign. We define for v ∈ L 1 (Ω), v + = max(v, 0) and v -= max(-v, 0). As observed already in [START_REF]Existence and multiplicity for an elliptic problem with critical growth in the gradient and signchanging coefficients[END_REF], the structure of the solution set depends on the size of the positive hump (i.e. c + λ ) but it is not affect by the size of the negative hump (i.e. c - λ ). Hoping to clarify this, we now write c λ under the form c λ = λc +c -and consider the problem

-∆u = (λc + (x) -c -(x))u + µ(x)|∇u| 2 + h(x) , u ∈ H 1 0 (Ω) ∩ L ∞ (Ω) , (P λ )
under the assumption

                     Ω ⊂ R N , N ≥ 2, is a bounded domain with boundary ∂Ω of class C 1,1 , c + , c -, h + ∈ L q (Ω) for some q > N /2 , µ, h -∈ L ∞ (Ω) , c + (x) ≥ 0, c -(x) ≥ 0 and c -(x)c + (x) = 0 a.e. in Ω, |Ω + | > 0,
where

Ω + := Supp(c + )
there exists a ε > 0 such that µ(x) ≥ µ 1 > 0 and c -= 0 in {x ∈ Ω : d(x, Ω + ) < ε}.

(A 1 )
For a definition of Supp(f ) with f ∈ L p (Ω), for some p ≥ 1, we refer to [START_REF] Brezis | Functional analysis, Sobolev spaces and partial differential equations[END_REF]Proposition 4.17]. Note also that the condition that c -= 0 on {x ∈ Ω : d(x, Ω + ) < ε} for some ε > 0, is reminiscent of the so-called "thick zero set" condition first introduced in [START_REF] Alama | On semilinear elliptic equations with indefinite nonlinearities[END_REF].

We also observe that, under the regularity assumptions of condition (A 1 ), any solution to (P λ ) belongs to C 0,τ (Ω) for some τ > 0. This can be deduce from [82, Theorem IX-2.2], see also [START_REF] Arcoya | Remarks on the uniqueness for quasilinear elliptic equations with quadratic growth conditions[END_REF]Proposition 2.1].

As in [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF][START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF][START_REF] Souplet | A priori estimates and bifurcation of solutions for an elliptic equation with semidefinite critical growth in the gradient[END_REF] we obtain our results using a topological approach, relying thus on the derivation of a priori bounds. In that direction our main result is the following.

Theorem 2.1.1. Assume (A 1 ). Then, for any

Λ 2 > Λ 1 > 0, there exists a constant M > 0 such that, for each λ ∈ [Λ 1 , Λ 2 ], any solution to (P λ ) satisfies sup Ω u ≤ M.
Having at hand this a priori bound, following the strategy of [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF], we show the existence of a continuum of solutions to (P λ ). More precisely, defining

Σ := {(λ, u) ∈ R × C(Ω) : u solves (P λ )}, (2.1.1)
we prove the following theorem.

Theorem 2.1.2. Assume (A 1 ) and suppose that (P 0 ) has a solution u 0 with c + u 0 0. Then, there exists a continuum C ⊂ Σ such that the projection of C on the λ-axis is an unbounded interval (-∞, λ] for some λ ∈ (0, +∞) and C bifurcates from infinity to the right of the axis λ = 0. Moreover:

1) for all λ ≤ 0, the problem (P λ ) has an unique solution u λ and this solution satisfies

u 0 -u 0 ∞ ≤ u λ ≤ u 0 .
2) there exists λ 0 ∈ (0, λ] such that, for all λ ∈ (0, λ 0 ), the problem (P λ ) has at least two solutions with u i ≥ u 0 for i = 1, 2. b) Note that problem (P 0 ) is given by

-∆u = -c -(x)u + µ(x)|∇u| 2 + h(x) , u ∈ H 1 0 (Ω) ∩ L ∞ (Ω) .
In [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF][START_REF] Coster | Existence and multiplicity for elliptic p-Laplacian problems with critical growth in the gradient[END_REF] the authors give sufficient conditions to ensure the existence of a solution to (P 0 ). Moreover, if h ≥ 0 in Ω, [17, Lemma 2.2] implies that the solution to (P 0 ) is non-negative.

Let us give some ideas of the proofs. As we do not have global sign conditions, the approaches used in [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF][START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF][START_REF] Souplet | A priori estimates and bifurcation of solutions for an elliptic equation with semidefinite critical growth in the gradient[END_REF] to obtain the a priori bounds do not apply anymore and another strategy is required. To this aim, we further develop some techniques first sketched in the unpublished work [START_REF]Uniform bounds via regularity estimates for elliptic PDE with critical growth in the gradient[END_REF]. These techniques, in the framework of viscosity solutions to fully nonlinear equations, now lies at the heart of the paper [START_REF] Nornberg | A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient[END_REF]. We also make use of some ideas from [START_REF] García-Meli Án | A priori bounds and existence of solutions for slightly superlinear elliptic problems[END_REF]. First we show, in Lemma 2.4.1, that it is sufficient to control the behavior of the solutions on Ω + . By compactness, we are then reduced to study what happens around an (unknown) point x ∈ Ω + . We shall consider separately the alternative cases x ∈ Ω + ∩Ω and x ∈ Ω + ∩∂Ω. A local analysis is made respectively in a ball or a semiball centered at x. If similar analysis, based on the use of Harnack type inequalities, had previously been performed in other contexts when x ∈ Ω, we believe it is not the case when x ∈ ∂Ω. For x ∈ ∂Ω, the key to our approach is the use of boundary weak Harnack inequality. Actually a major part of the chapter is devoted to establishing this inequality. This is done in a more general context than needed for (P λ ). In particular it also cover the case of the p-Laplacian with a zero order term. We believe that this "boundary weak Harnack inequality", see Theorem 2.3.1, is of independent interest and will proved to be useful in other settings. Its proof uses ideas introduced by B. Sirakov [START_REF]Boundary Harnack estimates and quantitative strong maximum principles for uniformly elliptic PDE[END_REF]. In [START_REF]Boundary Harnack estimates and quantitative strong maximum principles for uniformly elliptic PDE[END_REF] such type of inequalities is established for an uniformly elliptic operator and viscosity solutions. However, since our context is quite different, the result of [START_REF]Boundary Harnack estimates and quantitative strong maximum principles for uniformly elliptic PDE[END_REF] does not apply to our situation and we need to work out an adapted proof.

We now describe the organization of the chapter. In Section 2.2, we present some preliminary results which are needed in the development of our proofs. In Section 2.3, we prove the boundary weak Harnack inequality for the p-Laplacian. The a priori bound, namely Theorem 2.1.1, is proved in Section 2.4. Finally Section 2.5 is devoted to the proof of Theorem 2.1.2.

Notation.

1) In R N , we use the notations |x| = x 2 1 + . . . + x 2 N and B R (y) = {x ∈ R N : |x -y| < R}.
2) We denote R + = (0, +∞), R -= (-∞, 0) and N = {1, 2, 3, . . .}.

3) For h 1 , h 2 ∈ L 1 (Ω) we write

• h 1 ≤ h 2 if h 1 (x) ≤ h 2 (x) for a.e. x ∈ Ω, • h 1 h 2 if h 1 ≤ h 2 and meas({x ∈ Ω : h 1 (x) < h 2 (x)}) > 0.

Preliminary results

In this section, we collect some results which will play an important role throughout the chapter. First of all, let us consider the boundary value problem

-∆u + H(x, u, ∇u) = f , u ∈ H 1 0 (ω) ∩ L ∞ (ω). ( 2

.2.1)

Here ω ⊂ R N is a bounded domain, f ∈ L 1 (ω) and

H : ω × R × R N → R is a Carathéodory function. Definition 2.2.1. We say that α ∈ H 1 (ω) ∩ L ∞ (ω) is a lower solution to (2.2.1) if α + ∈ H 1 0 (ω) and, for all ϕ ∈ H 1 0 (ω) ∩ L ∞ (ω) with ϕ ≥ 0, we have ω ∇α∇ϕ dx + ω H(x, α, ∇α)ϕ dx ≤ ω f (x)ϕ dx . Similarly, β ∈ H 1 (ω) ∩ L ∞ (ω) is an upper solution to (2.2.1) if β -∈ H 1 0 (ω) and, for all ϕ ∈ H 1 0 (ω) ∩ L ∞ (ω) with ϕ ≥ 0, we have ω ∇β∇ϕ dx + ω H(x, β, ∇β)ϕ dx ≥ ω f (x)ϕ dx .
Next, we consider the boundary value problem

-∆u + a(x)u = b(x) , u ∈ H 1 0 (ω) , (2.2.2) under the assumption ω ⊂ R N , N ≥ 2 , is a bounded domain, a, b ∈ L r (ω) for some r > N /2. ( 2

.2.3)

Step 1: There exists

c 1 = c 1 (R, y, ω, a ∞ ) > 0 such that u(x) d(x, ∂ω) ≥ c 1 B R (y) f (x) dx , ∀ x ∈ B R/2 (y) . (2.2.4)
Since f is non-negative, observe that u is a non-negative upper solution to

-∆u + a(x)u = 0 , u ∈ H 1 0 (ω) .
Hence, by Lemma 2.2.3, there exists a constant

c 2 = c 2 (R, a ∞ ) > 0 such that u(x) ≥ c 2 B R (y) u dx , ∀ x ∈ B R/2 (y) . (2.2.5)
Now, let us denote by ξ the solution to

       -∆ξ + a ∞ ξ = χ B R (y) , in ω , ξ = 0 , on ∂ω . ( 2
u dx = ω u -∆ξ + a ∞ ξ dx ≥ ω f (x) ξ dx ≥ c 3 ω f (x) d(x, ∂ω) dx ≥ c 3 R B R (y) f (x) dx .
Hence, substituting the above information in (2.2.5) we obtain for

c 4 = c 2 c 3 R u(x) ≥ c 4 B R (y) f (x) dx , ∀ x ∈ B R/2 (y) , (2.2.7)
from which, since ω ⊂ R N is bounded, (2.2.4) follows.

Step 2: There exists c 5 = c 5 (R, y, ω, a ∞ ) > 0 such that

u(x) d(x, ∂ω) ≥ c 5 B R (y) f (x) dx , ∀ x ∈ ω \ B R/2 (y). ( 2

.2.8)

Let w be the unique solution to

           -∆w + a ∞ w = 0 , in ω \ B R/2 (y) , w = 0 , on ∂ω , w = 1 , on ∂B R/2 (y) .
(2.2.9)

Still by [29, Theorem 3], there exists c 6 = c 6 (R, y, ω, a ∞ ) > 0 such that w(x) ≥ c 6 d(x, ∂ω) for all x ∈ ω \ B R/2 (y). On the other hand, let us introduce

v(x) = u(x) c 4 B R (y) f (x) dx ,
with c 4 given in (2.2.7). Observe that v is an upper solution to (2.2.9). Hence, by the standard comparison principle, it follows that v(x) ≥ w(x) for all x ∈ ω \ B R/2 (y) and (2.2.8) follows.

Step 3: Conclusion.

The result follows from (2.2.4) and (2.2.8).

Boundary weak Harnack inequality

In this section we present a boundary weak Harnack inequality that will be central in the proof of Theorem 2.1.1. As we believe this type of inequality has its own interest, we establish it in the more general framework of the p-Laplacian. Recalling that ∆ p u = div(|∇u| p-2 ∇u) for 1 < p < ∞, we introduce the boundary value problem

-∆ p u + a(x)|u| p-2 u = 0 , u ∈ W 1,p 0 (ω) . (2.3.1)
Let us also recall that u ∈ W 1,p (ω) is an upper solution to (2.3.1) if u -∈ W We then prove the following result.

Theorem 2.3.1. (Boundary Weak Harnack Inequality) Let ω ⊂ R N , N ≥ 2, be a bounded domain with boundary ∂ω of class C 1,1 and let a ∈ L ∞ (ω) be a non-negative function. Assume that u is a non-negative upper solution to (2.3.1) and let x 0 ∈ ∂ω. Then, there exist

R > 0, ε = ε(p, R, a ∞ , ω) > 0 and C = C(p, R, ε, a ∞ , ω) > 0 such that, for all R ∈ (0, R] , inf B R (x 0 )∩ω u(x) d(x, ∂ω) ≥ C B R (x 0 )∩ω u(x) d(x, ∂ω) ε dx 1/ε .
As already indicated, in the proof of Theorem 2.3.1 we shall make use of some ideas from [START_REF]Boundary Harnack estimates and quantitative strong maximum principles for uniformly elliptic PDE[END_REF].

Before going further, let us introduce some notation that we will be used throughout the section. We define

r := r(N , p) =          N (p -1) N -p if p < N , + ∞ if p ≥ N ,
and denote by Q ρ (y) the cube of center y and side of length ρ, i.e.

Q ρ (y) = {x ∈ R N : |x i -y i | < ρ/2 for i = 1, . . . , N }.
In case the center of the cube is ρe with e = (0, 0, . . . , 1/2), we use the notation

Q ρ = Q ρ (ρe).
Let us now introduce several auxiliary results that we shall need to prove Theorem 2.3.1. We begin recalling the following comparison principle for the p-Laplacian.

Lemma 2.3.2. [112, Lemma 3.1] Let ω ⊂ R N , N ≥ 2,
be a bounded domain and let a ∈ L ∞ (ω) be a non-negative function. Assume that u , v ∈ W 1,p (ω) satisfy (in a weak sense)

       -∆ p u + a(x)|u| p-2 u ≤ -∆ p v + a(x)|v| p-2 v , in ω , u ≤ v , on ∂ω .
Then, it follows that u ≤ v.

As a second ingredient, we need the weak Harnack inequality.

Theorem 2.3.3. [90, Theorem 3.13] Let ω ⊂ R N , N ≥ 2, be a bounded domain and let a ∈ L ∞ (ω) be a non-negative function. Assume that u ∈ W 1,p (ω) is a non-negative upper solution to

-∆ p u + a(x)|u| p-2 u = 0 , u ∈ W 1,p 0 (ω) ,
and let Q ρ (x 0 ) ⊂ ω. Then, for any σ , τ ∈ (0, 1) and γ ∈ (0, r), there exists

C = C(p, γ, σ , τ, ρ, a ∞ ) > 0 such that inf Q τρ (x 0 ) u ≥ C Q σ ρ (x 0 ) u γ dx 1/γ .
In the next result, we deduce a more precise information on the dependence of C with respect to ρ. This is closely related to [START_REF] Trudinger | On Harnack type inequalities and their application to quasilinear elliptic equations[END_REF]Theorem 1.2] where however the constant still depends on ρ.

Corollary 2.3.4. Let a be a non-negative constant and γ ∈ (0, r). There exists C = C(p, γ, a) > 0 such that, for all 0 < ρ ≤ 1, any u ∈ W 1,p (Q 3 ρ 2 (e)) non-negative upper solution to

-∆ p u + a|u| p-2 u = 0 , u ∈ W 1,p 0 (Q 3 ρ 2 (e)), (2.3.2) satisfies inf Q ρ (e) u ≥ C ρ -N /γ Q ρ (e) u γ dx 1/γ .
Proof. Let C = C(p, a, γ) > 0 be the constant given by Theorem 2.3.3 applied with ρ = 3 2 and σ = τ = 2 3 . This means that if v ∈ W 1,p (Q 3 2 (e)) is a non-negative upper solution to

-∆ p v + a|v| p-2 v = 0 , v ∈ W 1,p 0 (Q 3 2 (e)), (2.3.3) then inf Q 1 (e) v(y) ≥ C Q 1 (e) v γ dy 1/γ .
As 0 < ρ ≤ 1, observe that if u is a non-negative upper solution to (2.3.2), then v defined by v(y) = u( ρy , ρ(y N -1 2 ) + 1 2 ), where y = (y , y N ) with y ∈ R N -1 , is a non-negative upper solution to (2.3.3). Thus, we can conclude that inf

Q ρ (e) u(x) = inf Q 1 (e) v(y) ≥ C Q 1 (e) v γ dy 1/γ = C ρ -N /γ Q ρ (e) u γ dx 1/γ .
Finally, we introduce a technical result of measure theory.

Lemma 2.3.5. [71, Lemma 2.1] Let E ⊂ F ⊂ Q 1 be
two open sets. Assume there exists α ∈ (0, 1) such that:

• |E| ≤ (1 -α)|Q 1 |. • For any cube Q ⊂ Q 1 , |Q ∩ E| ≥ (1 -α)|Q| implies Q ⊂ F.
Then, it follows that |E| ≤ (1cα)|F| for some constant c = c(N ) ∈ (0, 1). Now, we can perform the proof of the main result. We prove the boundary weak Harnack inequality for cubes and as consequence we obtain the desired result. Lemma 2.3.6 (Growth lemma). Let a be a non-negative constant. Given ν > 0, there exists k

= k(p, ν, a) > 0 such that, if u ∈ W 1,p (Q 3 2
) is a non-negative upper solution to

-∆ p u + a |u| p-2 u = 0 , u ∈ W 1,p 0 (Q 3 2 ) ,
and the following inequality holds

|{x ∈ Q 1 : u(x) > x N }| ≥ ν.
(2.3.4)

Then u(x) > kx N in Q 1 .
Remark 2.3.1. Before we prove the Lemma, observe that there is no loss of generality in considering

a a non-negative constant instead of a ∈ L ∞ (Q 3 2 ) non-negative. If u ≥ 0 satisfies -∆ p u + a(x)|u| p-2 u ≥ 0 , in Q 3 2 , then u satisfies also -∆ p u + a ∞ |u| p-2 u ≥ 0 , in Q 3 2 . Proof. Let us define S δ = Q 3 2 \ Q 3 2 -δ 3
2 e and fix c 1 = c 1 (ν) ∈ (0, 1 2 ) small enough in order to ensure that |S δ | ≤ ν 2 for any 0 < δ ≤ c 1 .

Step 1: For all δ ∈ (0,

c 1 ], it follows that |{x ∈ Q 3 2 -δ 3 2 e : u(x) > x N }| ≥ ν 2 .
Directly observe that

{x ∈ Q 1 : u(x) > x N } ⊂ {x ∈ Q 3 2 : u(x) > x N } ⊂ {x ∈ Q 3 2 -δ 3 2 e : u(x) > x N } ∪ S δ .
Hence, Step 1 follows from (2.3.4) and the choice of c 1 .

Step 2: For any ε > 0 and all δ ∈ (0, c 1 ], the following inequality holds

Q 3 2 -δ 3 2 e u ε dx 1/ε ≥ δ 2 ν 2 1/ε . (2.3.5)
Since u ≥ 0 and, for any

x ∈ Q 3 2 -δ 3 2 e we have x N ≥ δ 2 , it follows that Q 3 2 -δ ( 3 2 e) u ε dx ≥ {x∈Q 3 2 -δ ( 3 2 e): u(x)≥x N } u ε dx ≥ {x∈Q 3 2 -δ ( 3 2 e):u(x)≥x N } δ 2 ε dx = δ 2 ε x ∈ Q 3 2 -δ 3 2 e : u(x) ≥ x N .
Step 2 follows then from Step 1.

Step 3: For any ε ∈ (0, r) and all δ ∈ (0, c 1 ], there exists

C δ = C δ (p, ε, δ, a) > 0 such that inf Q 3 2 -δ 3 2 e u(x) x N ≥ δ C δ 3 ν 2 1/ε
. By Theorem 2.3.3 applied with ρ = 3 2 , x 0 = 3 2 e and τ = σ = 1 -2 3 δ, there exists a constant

C δ = C δ (p, ε, δ, a) > 0 such that inf Q 3 2 -δ 3 2 e u(x) ≥ C δ Q 3 2 -δ 3 2 e u ε dx 1/ε . Since for all x ∈ Q 3 2 -δ 3 2 e we have x N ≤ 3 2 ,
Step 3 follows from the above inequality and Step 2.

Step 4: Conclusion.

We fix ε ∈ (0, r), define k δ = δ C δ 3 ν 2 1/ε and introduce η : [-3-2c 1 4 , 3-2c 1 4 ] N -1 → R a C ∞ function satisfying η(x 1 , . . . , x N -1 ) =          0, if (x 1 , . . . , x N -1 ) ∈ -1 2 , 1 2
N -1 , c 1 2 , if (x 1 , . . . , x N -1 ) ∈ ∂ R N -1 -3-2c 1 4 , 3-2c 1 4 N -1
,

and 0 ≤ η(x 1 , . . . , x N -1 ) ≤ c 1 2 for (x 1 , . . . , x N -1 ) ∈ -3-2c 1 4 , 3-2c 1 4 N -1
.

Moreover, we consider the auxiliary function

v δ (x 1 , . . . , x N ) = 1 δ x N -η(x 1 , . . . , x N -1 ) 2 + x N -η(x 1 , . . . , x N -1 )
defined in

ω δ = (x 1 , . . . , x N ) ∈ -3-2c 1 4 , 3-2c 1 4 N -1 × 0, c 1 2 : η(x 1 , . . . , x N -1 ) ≤ x N ≤ δ 2 .
Observe that, in ω δ , we have 0 ≤ x Nη(x 1 , . . . , x N -1 ) ≤ δ 2 . Hence, there exists c 2 = c 2 (p, ν, a) ∈ (0, c 1 ] such that, for all 0 < δ ≤ c 2 ,

-∆ p v δ + a|v δ | p-2 v δ ≤ - 2 δ (p -1) + 2 p-1 N -1 i=1 ∂ ∂x i N -1 i=1 ( ∂η ∂x i ) 2 + 1 p-2 2 ∂η ∂x i + 3a 4 δ ≤ 0 , in ω δ .
On the other hand, we define u δ = 2u k δ and immediately observe that

-∆ p u δ + a|u δ | p-2 u δ ≥ 0 , in ω δ .

Now, since by

Step 3, we have

u δ ≥ 2k δ k δ δ 2 = δ ≥ v δ , for x N = δ 2 , it follows that u δ ≥ v δ on ∂ω δ .
Then, applying Lemma 2.3.2, it follows that, for any δ ∈ (0, c 2 ], v δ ≤ u δ in ω δ . For δ = c 2 /2, we obtain in particular

u(x) ≥ 1 2 k c 2 2 v c 2 2 (x) = 1 2 k c 2 2 2 c 2 x 2 N + x N ≥ 1 2 k c 2 2 x N , in ω c 2 2 ∩ Q 1 .
The result then follows from the above inequality and Step 3.

Lemma 2.3.7. Let a ∈ L ∞ (Q 4 ) be a non-negative function. Assume that u ∈ W 1,p (Q 4 ) is a non-negative upper solution to -∆ p u + a(x)|u| p-2 u = 0 , u ∈ W 1,p 0 (Q 4 ) , (2.3.6) satisfying inf Q 1 u(x) x N ≤ 1 .
Then, there exist M = M(p, a ∞ ) > 1 and µ ∈ (0, 1) such that

{x ∈ Q 1 : u(x)/x N > M j } < (1 -µ) j , ∀ j ∈ N . (2.3.7)
Proof. Let us fix some notation that we use throughout the proof. We fix γ ∈ (0, r) and consider C 1 = C 1 (p, a ∞ ) > 0 the constant given by Corollary 2.3.4. We introduce α ∈ (0, 1) and fix C 2 ∈ (0, 1) the constant given by Lemma 2.3.5. Moreover, we choose ν = (1α) 1 4 N and denote by k = k(ν, p, a ∞ ) ∈ (0, 1) the constant given by Lemma 2.3.6 applied to an upper solution to

-∆ p u + 2 p a ∞ |u| p-2 u = 0 , u ∈ W 1,p 0 (Q 3 2 ) , (2.3.8)
with the chosen ν. Let us point out that, if u is a non-negative upper solution to (2.3.6), then u is a non-negative upper solution to (2.3.8). Finally, we consider

M ≥ max 1 k , 4 C 1 (1 -α) -1/γ ,
and we are going to show that (2.3.7) holds with µ = αC 2 .

First of all, observe that {x ∈ Q

1 : u(x)/x N > M} ⊂ {x ∈ Q 1 : ku(x) > x N }. Hence, since inf Q 1 ku(x)/x N ≤ k, Lemma 2.3.6 implies that |{x ∈ Q 1 : u(x)/x N > M}| ≤ |{x ∈ Q 1 : ku(x) > x N }| < ν < 1 -α < 1 -C 2 α (2.3.9)
and, in particular, (2.3.7) holds for j = 1. Now, let us introduce, for j ∈ N \ {1},

E = {x ∈ Q 1 : u(x)/x N > M j } and F = {x ∈ Q 1 : u(x)/x N > M j-1 } .
Since M > 1 and j ∈ N \ {1}, observe that (2.3.9) implies that

|E| = |{x ∈ Q 1 : u(x)/x N > M j }| ≤ |{x ∈ Q 1 : u(x)/x N > M}| ≤ 1 -α , (2.3.10)
and the first assumption of Lemma 2.3.5 is satisfied.

Claim: For every cube Q ρ (x 0 ) ⊂ Q 1 such that |E ∩ Q ρ (x 0 )| ≥ (1 -α)|Q ρ (x 0 )| = (1 -α)ρ N . (2.3.11)
we have Q ρ (x 0 ) ⊂ F.

Let us denote x 0 = (x 0 , x 0 N ) with x 0 ∈ R N -1 . We define the new variable y =

x -x 0 ρ , x N ρ , where ρ = 2x 0 N , and the rescaled function v(y) = 1 ρ u(ρ y + x 0 , ρ y N ). Then v is a non-negative upper solution to

-∆ p v + 2 p a ∞ |v| p-2 v = 0 , in Q 4/ρ -x 0 /ρ , 2/ρ . (2.3.12)
Moreover, observe that

x ∈ E ∩ Q ρ (x 0 ) if and only if y ∈ {y ∈ Q ρ/ρ (e) : v(y)/M j > y N } , and so, that (2.3.11) is equivalent to

|{y ∈ Q ρ/ρ (e) : v(y)/M j > y N }| ≥ (1 -α)|Q ρ/ρ (e)| = (1 -α) ρ ρ N . (2.3.13)
Observe also that the embedding

Q ρ (x 0 ) ⊂ Q 1 implies that ρ ≤ ρ ≤ 2 -ρ and |x 0,i | ≤ 1-ρ 2 for i ∈ {1, . . . , N -1}. In particular, we have Q 3 2 ⊂ Q 4/ρ -x 0 /ρ , 2/ρ .
|{y ∈ Q 1 : v(y)/M j > y N }| ≥ |{y ∈ Q ρ/ρ (e) : v(y)/M j > y N }| ≥ ν .
Hence, by Lemma 2.3.6, v(y)/M j > ky N in Q 1 and so, by the definition of k, v(y)/y N > M j-1 in Q ρ/ρ (e). This implies that u(x)/x N > M j-1 in Q ρ (x 0 ). Case 2: ρ < ρ /4. Recall that v/M j is a non-negative upper solution to (2.3.8). Hence, v/M j is also a non-negative upper solution to

-∆ p u + 2 p a ∞ |u| p-2 u = 0 , in Q 3ρ 2ρ (e) ⊂ Q 3 2 ,
Thus, by Corollary 2.3.4, we deduce that inf

Q ρ/ρ (e) v(y) M j ≥ C 1 ρ ρ -N Q ρ/ρ (e) v M j γ dy 1/γ . ( 2
|G| ≥ |{y ∈ Q ρ/ρ (e) : v(y)/M j > y N }| ≥ (1 -α) ρ ρ N .
Hence, we deduce that

Q ρ/ρ (e) v M j γ dy ≥ G v M j γ dy > 1 4 γ |G| ≥ 1 4 γ (1 -α) ρ ρ N ,
and so, by (2.3.14), that inf

Q ρ/ρ (e) v M j > C 1 4 (1 -α) 1/γ .
Finally, using that M ≥ 4 C 1

(1α) -1/γ and that y N ≤ 1 in Q ρ/ρ (e), we deduce that v(y) > M j-1 y N in Q ρ/ρ (e). Thus, we can conclude that u(x)/x N > M j-1 in Q ρ (x 0 ).

In both cases we prove that u(x)/x N > M j-1 in Q ρ (x 0 ). This means that Q ρ (x 0 ) ⊂ F and so, the Claim is proved. Since (2.3.10) and the Claim hold, we can apply Lemma 2.3.5 and we obtain that |E| ≤ (1 -

C 2 α)|F| , i.e. |{x ∈ Q 1 : u(x)/x N > M j }| ≤ (1 -C 2 α) |{x ∈ Q 1 : u(x)/x N > M j-1 }| , ∀ j ∈ N \ {1} .
Iterating in j and using (2.3.9), the result follows with µ = C 2 α ∈ (0, 1) depending only on N . Theorem 2.3.8 (Boundary weak Harnack inequality for cubes). Let a ∈ L ∞ (Q 4 ) be a non-negative function. Assume that u ∈ W 1,p (Q 4 ) is a non-negative upper solution to

-∆ p u + a(x)|u| p-2 u = 0 , u ∈ W 1,p 0 (Q 4 ), Then, there exist ε = ε(p, a ∞ ) > 0 and C = C(p, ε, a ∞ ) > 0 such that inf Q 1 u(x) x N ≥ C Q 1 u(x) x N ε dx 1/ε .
Proof. Let us split the proof into three steps.

Step 1:

Assume that inf Q 1 u(x) x N ≤ 1. Then, there exist ε = ε(p, a ∞ ) > 0 and C = C(p, ε, a ∞ ) > 0 such that, for all t ≥ 0, |{x ∈ Q 1 : u(x)/x N > t}| ≤ C min{1, t -2ε } .
Let us define the real valued function

f (t) = |{x ∈ Q 1 : u(x)/x N > t}| ,
and let M and µ be the constants obtained in Lemma 2.3.7. We define

C = max{(1 -µ) -1 , M 2ε } > 1 and ε = - 1 2 ln(1 -µ) ln M > 0 . If t ∈ [0, M], we easily get |{x ∈ Q 1 : u(x)/x N > t}| ≤ 1 ≤ CM -2ε ≤ C min{1, t -2ε }.
Hence, let us assume t > M > 1. Without loss of generality, we assume t ∈ [M j , M j+1 ] for some j ∈ N, and it follows that ln t ln M -1 ≤ j ≤ ln t ln M .

Since f is non-increasing and 1µ ∈ (0, 1), the above inequality and Lemma 2.3.7 imply

f (t) ≤ f (M j ) ≤ (1 -µ) j ≤ (1 -µ) ln t ln M -1 . (2.3.15)
Finally, observe that ln (1-µ)

ln t ln M -1 = ln t ln M -1 ln(1-µ) = ln t ln(1 -µ) ln M -ln(1-µ) ≤ -2ε ln t +ln C = ln(Ct -2ε ) . (2.3.16)
The Step 1 then follows from (2.3.15), (2.3.16) and the fact that min{1, t -2ε } = t -2ε for t ≥ 1.

Step 2:

Assume that inf Q 1 u(x) x N ≤ 1. Then, there exists C = C(p, ε, a ∞ ) > 0 such that Q 1 u(x) x N ε dx ≤ C < +∞ . (2.3.17)
Directly, applying [66, Lemma 9.7], we obtain that

Q 1 u(x) x N ε dx = ε ∞ 0 t ε-1 |{x ∈ Q 1 : u(x)/x N > t}| dt .
Hence, (2.3.17) follows from Step 1.

Step 3: Conclusion.

Let us introduce the function

v = u inf y∈Q 1 u(y) y N + β ,
where β > 0 is an arbitrary positive constant. Obviously, v satisfies the hypothesis of Step 2. Hence, applying Step 2, we obtain that

Q 1 u(x) x N ε 1 inf y∈Q 1 u(y) y N + β ε dx ≤ C , or equivalently that 1 C 1/ε Q 1 u(x) x N ε dx 1/ε ≤ inf Q 1 u(x) x N + β .
Letting β → 0 we obtain the desired result.

Proof of Theorem 2.3.1. Thanks to the regularity of the boundary, there exists R > 0 and a diffeomorphism ϕ such that ϕ(B R (x 0 ) ∩ ω) ⊂ Q 1 and ϕ(B R (x 0 ) ∩ ∂ω) ⊂ {x ∈ ∂Q 1 : x N = 0}. The result then follows from Theorem 2.3.8.

We end this section by presenting a corollary of Theorem 2.3.1. Consider the equation

-∆u + a(x)u = b(x) , u ∈ H 1 0 (ω) , (2.3.18)
under the assumption

           ω ⊂ R N , N ≥ 2, is a bounded domain with boundary ∂ω of class C 1,1 , a ∈ L ∞ (ω) , b -∈ L p (ω) for some p > N and b + ∈ L 1 (ω) ,
a ≥ 0 a.e. in ω .

(

, ε = ε(R, a ∞ , ω) > 0, C 1 = C 1 (R, ε, a ∞ , ω) > 0 and C 2 = C 2 (ω, a ∞ ) > 0 such that, for all R ∈ (0, R], inf B R (x 0 )∩ω u(x) d(x, ∂ω) ≥ C 1 B R (x 0 )∩ω u(x) d(x, ∂ω) ε dx 1/ε -C 2 b - L p (ω) .
In order to prove Corollary 2.3.9 we need the following lemma Lemma 2.3.10. Let ω ⊂ R N , N ≥ 2, be a bounded domain with boundary ∂ω of class C 1,1 and let a ∈ L ∞ (ω) and g ∈ L p (ω), p > N , be non-negative functions. Assume that u ∈ H 1 (ω) is a lower solution to -∆u + a(x)u = g(x) , u ∈ H 1 0 (ω) . Then there exists

C = C(ω, a ∞ ) > 0 such that sup ω u(x) d(x, ∂ω) ≤ C g L p (ω) .
Proof. First of all, observe that it is enough to prove the result for v solution to

-∆v + a(x)v = g(x) , in ω , v = 0 , on ∂ω .
as, by the standard comparison principle it follows that u ≤ v. Applying [66, Theorem 9.15 and Lemma 9.17] we deduce that v ∈ W 2,p 0 (ω) and there exists

C 1 = C 1 (ω, a ∞ ) > 0 such that v W 2,p (ω) ≤ C 1 g L p (ω) .
Moreover, as p > N , by Sobolev's inequality, we have

C 2 = C 2 (ω, a ∞ ) with v C 1 (ω) ≤ C 2 g L p (ω) ,
and so, we easily deduce that

v(x) ≤ C 3 g L p (ω) d(x, ∂ω) , ∀ x ∈ ω .
Hence, since u ≤ v, the result follows from the above inequality.

Proof of Corollary 2.3.9. Let w ≥ 0 be the solution to

-∆w + a(x)w = b -(x) , in ω , w = 0 , on ∂ω . (2.3.20) Observe that v = u + w satisfies -∆v + a(x)v ≥ 0, in ω , v ≥ 0 , on ∂ω . (2.3.21) Hence, by Theorem 2.3.1, there exist R > 0, ε = ε(p, R, a ∞ , ω) > 0 and C = C(p, R, ε, a ∞ , ω) > 0 such that, for all R ∈ (0, R] , inf B R (x 0 )∩ω v(x) d(x, ∂ω) ≥ C B R (x 0 )∩ω v(x) d(x, ∂ω) ε dx 1/ε . (2.3.22)
On the other hand, by Lemma 2.3.10, there exists

C 2 = C 2 (ω, a ∞ ) > 0 such that sup ω w(x) d(x, ∂ω) ≤ C 2 b - L p (ω) . ( 2
Ω + u + , in Ω.
Step 2: u ≥sup Ω + u --M.

We now define v = u + sup ∂D u -and obtain v ≥ 0 on ∂D as well as

-∆v = -c -(x)v + µ(x)|∇v| 2 + h(x) + c -(x) sup ∂D u -≥ -c -(x)v + µ(x)|∇v| 2 + h(x) , in D .
Thus v is an upper solution to (2.4.2). Now defining ṽ = ũũ ∞ , again, we have ṽ ≤ 0 on ∂D as well as

-∆ ṽ = -c -(x) ṽ + µ(x)|∇ ṽ| 2 + h(x) -c -(x) ũ ∞ ≤ -c -(x) ṽ + µ(x)|∇ ṽ| 2 + h(x) , in D .
Thus ṽ is a lower solution to (2.4.2). As previously we have that v, ṽ ∈ H

1 (D) ∩ W 1,N loc (D) ∩ C(D) and applying [17, Lemma 2.2] we obtain that ṽ ≤ v in D. Namely ũ -ũ ∞ ≤ u + sup ∂D u -, in D. Thus u ≥ ũ -ũ ∞ -sup ∂D u -, in D,
and without restriction we get that u ≥sup

Ω + u --M , in Ω,
ending the proof. Now, let u ∈ H 1 0 (Ω) ∩ L ∞ (Ω) be a solution to (P λ ). Following [18, Proposition 6.1], we introduce

w i (x) = 1 µ i e µ i u(x) -1 and g i (s) = 1 µ i ln(1 + µ i s), i = 1, 2 , (2.4.3)
where µ 1 is given in (A 1 ) and µ 2 = esssup µ(x). Observe that

u = g i (w i ) and 1 + µ i w i = e µ i u , i = 1, 2 ,
and that, by standard computations,

-∆w i = (1 + µ i w i ) (λc + (x) -c -(x))g i (w i ) + h(x) + e µ i u |∇u| 2 (µ(x) -µ i ). (2.4.4)
Using (2.4.4) we shall obtain a uniform a priori upper bound on u in a neighborhood of any fixed point x ∈ Ω + . We consider the two cases x ∈ Ω + ∩ Ω and x ∈ Ω + ∩ ∂Ω separately. Lemma 2.4.2. Assume that (A 1 ) holds and that x ∈ Ω + ∩ Ω. For each Λ 2 > Λ 1 > 0, there exist M I > 0 and R > 0 such that, for any λ ∈ [Λ 1 , Λ 2 ], any solution u of (P λ ) satisfies sup B R (x) u ≤ M I .

Proof. Under the assumption (A 1 ) we can find a R > 0 such that µ(x) ≥ µ 1 > 0, c -≡ 0 in B 4R (x) ⊂ Ω and c + 0 in B R (x). For simplicity, in this proof, we denote B mR = B mR (x), for m ∈ N.

Since c -≡ 0 and µ(x) ≥ µ 1 in B 4R , observe that (2.4.4) reduces to

-∆w 1 + µ 1 h -(x)w 1 ≥ λ(1 + µ 1 w 1 )c + (x)g 1 (w 1 ) + h + (x)(1 + µ 1 w 1 ) -h -(x) , in B 4R .
(2.4.5)

Let z 2 be the solution to

-∆z 2 + µ 1 h -(x)z 2 = -Λ 2 c + (x) e -1 µ 1 , z 2 ∈ H 1 0 (B 4R ). ( 2
= D(x, µ 1 , Λ 2 , h - L q (B 4R ) , c + L q (B 4R ) , q, R) > 0 such that z 2 ≥ -D in B 4R . (2.4.7)
Moreover, by the weak maximum principle [66, Theorem 8.1], we have that z 2 ≤ 0. Now defining

v 1 = w 1 -z 2 + 1 µ 1
, and since min

[-1/µ i ,+∞[ (1 + µ i s)g i (s) = -e -1
µ i

, we observe that v 1 satisfies

-∆v 1 + µ 1 h -(x)v 1 ≥ Λ 1 c + (x)(1 + µ 1 w 1 )g 1 (w 1 ) + , in B 4R . (2.4.8) Also, since w 1 > -1/µ 1 , we have v 1 > 0 in B 4R . Note also that 0 < 1 + µ 1 w 1 = µ 1 v 1 + µ 1 z 2 in B 4R .
Now, we split the rest of the proof into four steps.

Step 1: There exist

C 1 = C 1 (x, Λ 1 , Λ 2 , R, µ 1 , q, h - L ∞ (B 4R ) , c + L q (B 4R ) ) > 0 such that k := inf B R v 1 (x) ≤ C 1 .
(2.4.9)

In case µ 1 inf B R v 1 (x) ≤ 1 + µ 1 D
, where D is given by (2.4.7), the Step 1 is proved. Hence, we assume that

µ 1 v 1 (x) ≥ 1 + µ 1 D, ∀ x ∈ B R .
(2.4.10)

In particular,

µ 1 v 1 + µ 1 z 2 ≥ 1 on B R . Now,
C = C(R, h - L ∞ (B 4R ) , µ 1 , Λ 1 , x) > 0 such that, k ≥ C B R c + (y) µ 1 v 1 (y) + µ 1 z 2 (y) ln µ 1 v 1 (y) + µ 1 z 2 (y) dy ≥ C B R c + (y) (µ 1 k -µ 1 D) ln µ 1 k -µ 1 D dy = C(µ 1 k -µ 1 D) ln µ 1 k -µ 1 D c + L 1 (B R ) .
As c + 0 in B R , comparing the growth in k of the various terms, we deduce that k must remain bounded and thus the existence of

C 1 = (x, Λ 1 , Λ 2 , R, µ 1 , q, h - L ∞ (B 4R ) , c + L q (B 4R
) ) > 0 such that (2.4.9) holds.

Step 2: For any 1 ≤ s < N N -2 , there exists

C 2 = C 2 (x, µ 1 , R, s, Λ 1 , Λ 2 , q, h - L ∞ (B 4R ) , c + L q (B 4R ) ) > 0 such that B 2R (1 + µ 1 w 1 ) s dx ≤ C 2 .
Applying Lemma 2.2.3 to (2.4.8), we deduce the existence of

C = C(s, µ 1 , R, h - L q (B 4R ) ) > 0 such that B 2R v s 1 dx 1/s ≤ C inf B R v 1 .
The Step 2 follows from Step 1 observing that 0

≤ 1 + µ 1 w 1 = µ 1 v 1 + µ 1 z 2 ≤ µ 1 v 1 .
Step 3: For any 1 ≤ s < N N -2 , we have, for the constant

C 2 > 0 introduced in Step 2, that B 2R 1 + µ 2 w 2 µ 1 s µ 2 dx ≤ C 2 .
This directly follows from Step 2 since, by the definition of w i , we have

(1 + µ 2 w 2 ) µ 1 µ 2 = (e µ 2 u ) µ 1 µ 2 = e µ 1 u = (1 + µ 1 w 1 ).
Step 4: Conclusion.

We will show the existence of

C 3 = C 3 (x, µ 1 , µ 2 , R, Λ 1 , Λ 2 , q, h - L ∞ (B 4R ) , c + L q (B 4R ) ) > 0 such that sup B R w 2 ≤ C 3 .
Thus, thanks to the definition of w 2 , we can conclude the proof. Let us fix s ∈ [1, N N -2 ), r ∈ ( N 2 , q) and α = (q-r)µ 1 s µ 2 qr and let c α > 0 such that

ln(1 + x) ≤ (1 + x) α + c α , ∀ x ≥ 0.
We introduce the auxiliary functions

a(x) = Λ 2 c + (x)(1 + µ 2 w 2 ) α + c α Λ 2 c + (x) + µ 2 h + (x), b(x) = Λ 2 µ 2 c + (x)(1 + µ 2 w 2 ) α + c α Λ 2 µ 2 c + (x) + h + (x) + c -(x) e -1 µ 2 ,
and, as µ(x) ≤ µ 2 , we deduce from (2.4.4) that w 2 satisfies

-∆w 2 ≤ a(x)w 2 + b(x)
in Ω , w 2 = 0 on ∂Ω. Now, as q/r > 1, by Step 3 and Hölder inequality, it follows that

B 2R (c + (x)(1 + µ 2 w 2 ) α ) r dx ≤ c + r L q (B 2R ) B 2R (1 + µ 2 w 2 ) αqr q-r dx q-r q ≤ c + r L q (B 2R ) B 2R (1 + µ 2 w 2 ) µ 1 s µ 2 dx q-r q ≤ C q-r q 2 c + r L q (B 2R ) .
Hence, there exists

D(x, µ 1 , µ 2 , s, R, Λ 1 , Λ 2 , q, h - L ∞ (B 4R ) , c + L q (B 4R ) , r, h + L q (B 2R ) ) > 0 such that max{ a L r (B 2R ) , b L r (B 2R ) } ≤ D . (2.4.11)
Applying then Lemma 2.2.1, there exists

C(x, µ 1 , µ 2 , s, R, Λ 1 , Λ 2 , q, h - L q (B 4R ) , c + L q (B 4R ) ) > 0 such that sup B R w + 2 ≤ C B 2R (w + 2 ) µ 1 µ 2 s dx µ 2 µ 1 s + b L r (B 2R ) ≤ C B 2R (w + 2 ) µ 1 µ 2 s dx µ 2 µ 1 s + D .
On the other hand, by Step 3, we get

B 2R (w + 2 ) µ 1 µ 2 s dx ≤ C(µ 1 , µ 2 , s) B 2R (1 + µ 2 w 2 ) µ 1 µ 2 s dx ≤ C(µ 1 , µ 2 , s) C 2 ,
and the result follows.

Lemma 2.4.3. Assume that (A 1 ) holds and that x ∈ Ω + ∩ ∂Ω. For each Λ 2 > Λ 1 > 0, there exist R > 0 and M B > 0 such that, for any λ ∈ [Λ 1 , Λ 2 ], any solution to

(P λ ) satisfies sup B R (x)∩Ω u ≤ M B .
Proof. Let R > 0 given by Theorem 2.3.1. Under the assumption (A 1 ), we can find R ∈ (0, R/2] and

Ω 1 ⊂ Ω with ∂Ω 1 of class C 1,1 such that B 2R (x) ∩ Ω ⊂ Ω 1 and µ(x) ≥ µ 1 > 0, c -≡ 0 and c + 0 in Ω 1 .
Since c -≡ 0 and µ(x) ≥ µ 1 in Ω 1 , observe that (2.4.4) reduces to

-∆w 1 + µ 1 h -(x)w 1 ≥ λ(1 + µ 1 w 1 )c + (x)g 1 (w 1 ) + h + (x)(1 + µ 1 w 1 ) -h -(x) , in Ω 1 (2.4.12)
Let z 2 be the solution to

-∆z 2 + µ 1 h -(x)z 2 = -Λ 2 c + (x) e -1 µ 1 , z 2 ∈ H 1 0 (Ω 1 ). (2.4.13)
As in Lemma 2.4.2, z 2 ∈ C(Ω 1 ) and there exists a

D = D(µ 1 , Λ 2 , h - L q (B 4R ) , c + L q (B 4R ) , q, Ω 1 ) > 0 such that -D ≤ z 2 ≤ 0 on Ω 1 . Now defining v 1 = w 1 -z 2 + 1 µ 1 we observe that v 1 satisfies -∆v 1 + µ 1 h -(x)v 1 ≥ Λ 1 c + (x)(1 + µ 1 w 1 )g 1 (w 1 ) + , in Ω 1 .
(2.4.14)

and v 1 > 0 on Ω 1 . Note also that 0

< 1 + µ 1 w 1 = µ 1 v 1 + µ 1 z 2 on Ω 1 .
Next, we split the rest of the proof into three steps.

Step 1: There exists

C 1 = C 1 (Ω 1 , x, Λ 1 , Λ 2 , R, µ 1 , q, h - L ∞ (Ω 1 ) , c + L q (Ω 1 ) ) > 0 such that inf B 2R (x)∩Ω 1 v 1 (x) d(x, ∂Ω 1 ) ≤ C 1 . Choose R 2 > 0 and y ∈ Ω such that B 4R 2 (y) ⊂ B 2R (x) ∩ Ω and c + 0 in B R 2 (y). As in Step 1 of Lemma 2.4.2, there exists C = C(Ω 1 , y, Λ 1 , Λ 2 , R 2 , µ 1 , q, h - L ∞ (Ω 1 ) , c + L q (Ω 1 ) ) > 0 such that inf B R 2 (y) v 1 (x) ≤ C .
We conclude by observing, since

B 4R 2 (y) ⊂ B 2R (x) ∩ Ω 1 , that inf B 2R (x)∩Ω 1 v 1 (x) d(x, ∂Ω 1 ) ≤ inf B R 2 (y) v 1 (x) d(x, ∂Ω 1 ) ≤ 1 3R 2 inf B R 2 (y) v 1 (

x).

Step 2:

There exist ε = ε(R, µ 1 , h - L ∞ (Ω 1 ) , Ω 1 ) > 0 and C 2 = C 2 (x, µ 1 , R, R, s, Λ 1 , Λ 2 , q, h - L ∞ (Ω 1 ) , c + L q (Ω 1 ) ) > 0 such that B 2R (x)∩Ω (1 + µ 1 w 1 ) ε dx 1/ε ≤ C 2 .

By Theorem 2.3.1 applied on (2.4.14) and Step 1, we obtain constants

ε = ε(R, µ 1 , h - L ∞ (Ω 1 ) , Ω 1 ) > 0 and C = C(Ω 1 , x, µ 1 , ε, R, Λ 1 , Λ 2 , q, h - L ∞ (Ω 1 ) , c + L q (Ω 1 ) ) > 0 such that B 2R (x)∩Ω 1 v 1 (x) d(x, ∂Ω 1 ) ε dx 1/ε ≤ C . This clearly implies, since Ω 1 ⊂ Ω, that B 2R (x)∩Ω 1 v 1 (x) ε dx 1/ε ≤ C diam(Ω) .
The Step 2 then follows observing that 0

≤ 1 + µ 1 w 1 = µ 1 v 1 + µ 1 z 2 ≤ µ 1 v 1 and taking into account that B 2R (x) ∩ Ω = B 2R (x) ∩ Ω 1 .
Step 3: Conclusion.

Arguing exactly as in Step 3 and 4 of Lemma 2.4.2, using Lemma 2.2.2 and Step 2, we show the existence of

C 3 = C 3 (x, µ 1 , µ 2 , R, Λ 1 , Λ 2 , h - L ∞ (Ω 1 ) , c + L q (B 2R (Ω 1 ) ) > 0 such that sup B R (x)∩Ω w 2 ≤ C 3 .
Hence, the proof of the lemma follows by the definition of w 2 .

Proof of Theorem 2.1.1. Arguing by contradiction we assume the existence of sequences

{λ n } ⊂ [Λ 1 , Λ 2 ]
, {u n } solutions to (P λ ) for λ = λ n and of points {x n } ⊂ Ω such that

u n (x n ) = max{u n (x) : x ∈ Ω} → ∞ , as n → ∞ . (2.4.15)
Observe that Lemma 2.4.1 and (2.4.15) together imply the existence of a sequence of points

y n ∈ Ω + such that u n (y n ) = max{u n (y) : y ∈ Ω + } → ∞ , as n → ∞ . (2.4.16)
Passing to a subsequence if necessary, we may assume that λ n → λ ∈ [Λ 1 , Λ 2 ] and y n → y ∈ Ω + . Now, let us distinguish two cases:

• If y ∈ Ω + ∩ Ω, Lemma 2.4
.2 shows that we can find R I > 0 and

M I > 0 such that, if u ∈ H 1 0 (Ω) ∩ L ∞ (Ω) is a solution to (P λ ), then sup B R I (y) u ≤ M I .
This contradicts (2.4.16).

• If y ∈ Ω + ∩ ∂Ω, Lemma 2.4.3 shows that we can find R B > 0 and M B > 0 such that, if u ∈ H 1 0 (Ω) ∩ L ∞ (Ω) is a solution to (P λ ), then sup B R B (y)∩Ω u ≤ M B . Again, this contradicts (2.4.16).
As (2.4.16) cannot happen, the result follows.

Proof of Theorem 2.1.2

Let us begin with a preliminary result.

Lemma 2.5.1. Under the assumption (A 1 ), assume that (P 0 ) has a solution u 0 for which there exist x ∈ Ω and R > 0 such that c + u 0 0, c -≡ 0 and µ ≥ 0 in B R (x). Then there exists Λ ∈ (0, ∞) such that, for λ ≥ Λ, the problem (P λ ) has no solution u with u ≥ u 0 in B R (x).

Proof. Let us introduce c(x) := min{c + (x), 1}. Observe that 0 c ≤ c + and define γ 1 1 > 0 as the first eigenvalue of the problem

-∆ϕ = γc(x)ϕ in B R (x), ϕ = 0 on ∂B R (x). (2.5.1)
By standard arguments, there exists ϕ 1 1 ∈ C 1 0 (B R (x)) an associated first eigenfunction such that ϕ 1 1 (x) > 0 for all x ∈ B R (x) and, denoting by n the outward normal to ∂B R (x), we also have

∂ϕ 1 1 (x) ∂n < 0 , on ∂B R (x). (2.5.2)
Now, let us introduce the function φ ∈ H 1 0 (Ω) ∩ L ∞ (Ω), defined as

φ(x) = ϕ 1 1 (x) , x ∈ B R (x), 0 x ∈ Ω \ B R (x),
and suppose that u is a solution to (P λ ) such that u ≥ u 0 in B R (x). First observe that, in view of (2.5.2) and as u ≥ u 0 on B R (x), there exists a constant C > 0 independent of u such that

∂B R (x) u ∂ϕ 1 1 ∂n dS ≤ C. (2.5.3)
Thus on one hand, using (2.5.1) and (2.5.3), we obtain

Ω ∇φ∇u + c -(x) φ u dx = B R (x) ∇ϕ 1 1 ∇u dx = - B R (x) u∆ϕ 1 1 dx + ∂B R (x) u ∂ϕ 1 1 ∂n dS ≤ - B R (x) u∆ϕ 1 1 dx + C = γ 1 1 B R (x) c(x) ϕ 1 1 u dx + C ≤ γ 1 1 Ω c + (x) φ u dx + C.
(2.5.4)

On the other hand, considering φ as test function in (P λ ) we observe that

Ω ∇φ∇u + c -(x) φ u dx = λ Ω c + (x) u φ dx + Ω µ(x)|∇u| 2 + h(x) φ dx .
(2.5.5) From (2.5.4) and (2.5.5), we then deduce that, for a D > 0 independent of u.

(γ 1 1 -λ) Ω c + (x) φ u dx ≥ Ω µ(x)|∇u| 2 + h(x) φ dx -C = B R (x) µ(x)|∇u| 2 + h(x) ϕ 1 1 dx -C ≥ -D.
(2.5.6)

As c + u 0 0 in B R (x), we have that

Ω c + (x) φ u dx ≥ Ω c + (x) φ u 0 dx > 0.
Hence, for λ > γ 1 1 large enough, we obtain a contradiction with (2.5.6).

Proof of Theorem 2.1.2. We treat separately the cases λ ≤ 0 and λ > 0.

Part 1: λ ≤ 0.

Observe that for λ ≤ 0 we have λc +c -≤ -c -and hence the result follows from [18, Lemma 5.1, Proposition 4.1, Proposition 5.1, Theorem 2.2] as in the proof of [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF]Theorem 1.2]. Moreover, observe that u 0 is an upper solution to (P λ ). Hence we conclude that u λ ≤ u 0 by [17, Lemmas 2.1 and 2.2].

Part 2: λ > 0.

Consider, for λ ≥ 0 the modified problem

-∆u + u = (λc + (x) -c -(x) + 1) ((u -u 0 ) + + u 0 ) + µ(x)|∇u| 2 + h(x) , u ∈ H 1 0 (Ω) ∩ L ∞ (Ω). (P λ )
As in the case of (P λ ), any solution to (P λ ) belongs to C 0,τ (Ω) for some τ > 0. Moreover, observe that u is a solution to (P λ ) if and only if it is a fixed point of the operator T λ defined by T λ : C(Ω) → C(Ω) : v → u with u the solution to

-∆u + u -µ(x)|∇u| 2 = (λc + (x) -c -(x) + 1) ((v -u 0 ) + + u 0 ) + h(x) , u ∈ H 1 0 (Ω) ∩ L ∞ (Ω).
Applying [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF]Lemma 5.2], we see that T λ is completely continuous. Now, we denote

Σ := {(λ, u) ∈ R × C(Ω) : u solves (P λ )}
and we split the rest of the proof into three steps.

Step 1: If u is a solution to (P λ ) then u ≥ u 0 and hence it is a solution to (P λ ).

Observe that (uu 0 ) + + u 0u ≥ 0. Also we have that λc + (x)((uu 0 ) + + u 0 ) ≥ λc + (x)u 0 ≥ 0. Hence, we deduce that a solution u of (P λ ) is an upper solution to

-∆u = -c -(x) ((u -u 0 ) + + u 0 ) + µ(x)|∇u| 2 + h(x) , u ∈ H 1 0 (Ω) ∩ L ∞ (Ω).
(2.5.7)

Then the result follows from [17, Lemmas 2.1 and 2.2] noting that u 0 is a solution to (2.5.7).

Step 2: u 0 is the unique solution to (P 0 ) and i(I -T 0 , u 0 ) = 1.

Again the uniqueness of the solution to (P 0 ) can be deduced from [17, Lemmas 2.1 and 2.2]. Now, in order to prove that i(I -T 0 , u 0 ) = 1, we consider the operator S t defined by S t : C(Ω) → C(Ω) : v → u with u the solution to

-∆u + u -µ(x)|∇u| 2 = t[(-c -(x) + 1) (u 0 + (v -u 0 ) + -(v -u 0 -1) + ) + h(x)] , u ∈ H 1 0 (Ω) ∩ L ∞ (Ω).
First, observe that there exists R > 0 such that, for all t ∈ [0, 1] and all v ∈ C(Ω),

S t v ∞ < R. This implies that deg(I -S 1 , B(0, R)) = deg(I, B(0, R)) = 1.
By [17, Lemmas 2.1 and 2.2], we see that u 0 is the only fixed point of S 1 . Hence, by the excision property of the degree, for all ε > 0 small enough, it follows that

deg(I -S 1 , B(u 0 , ε)) = deg(I -S 1 , B(0, R)) = 1.
Thus, as for ε < 1, S 1 = T 0 , we conclude that

i(I -T 0 , u 0 ) = lim ε→0 deg(I -T 0 , B(u 0 , ε)) = lim ε→0 deg(I -S 1 , B(u 0 , ε)) = 1.
Step 3: Existence and behavior of the continuum.

By [99, Theorem 3.2] (see also [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF]Theorem 2.2]), there exists a continuum C ⊂ Σ such that C ∩ ([0, ∞) × C(Ω)) is unbounded. By Step 1, we know that if u ∈ C then u ≥ u 0 and is a solution to (P λ ). Thus applying Lemma 2.5.1, we deduce that Proj R C ∩ [0, ∞) ⊂ [0, Λ]. By Theorem 2.1.1 and

Step 1, we deduce that for every Λ 1 ∈ (0, Λ), there is an a priori bound on the solutions to

(P λ ) for λ ∈ [Λ 1 , Λ]. Hence, the projection of C ∩ ([Λ 1 , Λ) × C(Ω)) on C(Ω)
is bounded, and so, we deduce that C emanates from infinity to the right of λ = 0. Finally, since C contains (0, u 0 ) with u 0 the unique solution to (P 0 ), we conclude that there exists λ 0 ∈ (0, Λ) such that problem (P λ ), and thus problem (P λ ), has at least two solutions satisfying u ≥ u 0 for λ ∈ (0, λ 0 ).

3

Two solutions for an indefinite elliptic problem with critical growth in the gradient

Introduction and main results

In this chapter we will study the existence and multiplicity of solutions to boundary value problems of the form

-∆u = c(x)u + µ(x)|∇u| 2 + h(x), in Ω, u = 0, on ∂Ω, (P )
where Ω ⊂ R N , N ≥ 2, is a bounded domain with smooth boundary, c and h belong to L q (Ω) for some q > N /2, µ belongs to L ∞ (Ω) and the solutions are searched in H 1 0 (Ω) ∩ L ∞ (Ω). There exist several mathematical reasons that make the study of nonlinear elliptic PDEs with quadratic growth in the gradient interesting. For instance, J. L. Kazdan and R. J. Kramer observed in 1978 that second order PDEs with quadratic growth in the gradient are invariant under changes of variable of type v = F(u). This took them to claim on [78, page 619] that "In the long run, the class of semilinear equations should be less important than some more general class of equations that is invariant under changes of variables". From a pure mathematical point of view, it is also worth noting that, in Riemannian geometry, this type of equations naturally appears in the study of gradient Ricci solitons, see for instance [START_REF] Magliaro | On a paper of Berestycki-Hamel-Rossi and its relations to the weak maximum principle at infinity, with applications[END_REF]Section 1]. On the other hand, concerning more practical reasons, we would like to mention that problem (P ) with c ≡ 0 corresponds to the stationary case of the Kardar-Parisi-Zhang model of growing interfaces introduced in [START_REF] Kardar | Dynamic scaling of growing interfaces[END_REF].

The study of nonlinear elliptic PDEs with a gradient dependence up to the critical growth was essentially initiated by L. Boccardo, F. Murat and J.-P. Puel in the 80's. In the case where c(x) ≤ α 0 < 0 a.e. in Ω for some α 0 < 0, now referred to as the coercive case, the existence of a solution to (P ) is a particular case of the results of [START_REF] Boccardo | Existence de solutions faibles pour des équations elliptiques quasilinéaires à croissance quadratique[END_REF][START_REF]Quelques propriétés des opérateurs elliptiques quasi linéaires[END_REF][START_REF]L ∞ estimate for some nonlinear elliptic partial differential equations and application to an existence result[END_REF] and its uniqueness follows from [START_REF] Barles | Remarks on the maximum principle for nonlinear elliptic PDEs with quadratic growth conditions[END_REF][START_REF] Barles | Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions[END_REF]. The weakly coercive case c ≡ 0 was first studied in [START_REF] Ferone | Nonlinear problems having natural growth in the gradient: an existence result when the source terms are small[END_REF] where, for µh N /2 small enough, the authors proved the existence of a solution to (P ). For µ(x) ≡ µ > 0 constant and h 0 theses results were then improved in [4]. Finally, in the recent work [START_REF] Coster | Existence and multiplicity for elliptic p-Laplacian problems with critical growth in the gradient[END_REF] we completely characterised the existence of solutions to (P ) in the weakly coercive case c ≡ 0. The limit coercive case where one only requires c(x) ≤ 0 a.e. in Ω (i.e. allowing parts of the domain where c ≡ 0 and parts of it where c < 0) proved to be more complex to treat. In [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF], the authors observed that the existence of a solution to (P ) is not guaranteed and gave sufficient conditions to ensure such existence. In case h does not have a sign, weaker sufficient conditions can be found in [START_REF] Coster | Existence and multiplicity for elliptic p-Laplacian problems with critical growth in the gradient[END_REF]. The fact that the uniqueness also holds in the limit coercive case c ≤ 0 was proved in [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF]. We refer likewise to [START_REF] Arcoya | Remarks on the uniqueness for quasilinear elliptic equations with quadratic growth conditions[END_REF] for more general uniqueness results in this framework. Finally, let us point out that, except for [4], all these results were obtained without requiring any sign conditions on µ and h.

If c(x) 0 a.e. in Ω, i.e. c 0 or c changes sign, problem (P ) behaves very differently and becomes much more richer than for c ≤ 0. The first paper which addressed this situation was [START_REF] Jeanjean | Existence and multiplicity for elliptic problems with quadratic growth in the gradient[END_REF]. Following [START_REF] Sirakov | Solvability of uniformly elliptic fully nonlinear PDE[END_REF], which considered a particular case, the authors studied (P ) with c 0 and µ(x) ≡ µ > 0 constant. For c q and µh N /2 small enough the existence of two solutions to (P ) was obtained. This result has now been improved in several ways. The restriction µ constant was first removed in [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF]. In that paper the authors imposed on c a dependence on a real parameter λ and considered λc 0. For µ(x) ≥ µ 1 > 0 a.e. in Ω and h 0, they proved the existence of at least two solutions for λ > 0 small enough. In this direction we refer also to [START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF] where, imposing stronger regularity on c and h, the authors removed the condition h 0. Under different sets of assumptions, the authors clarified the structure of the set of solutions to (P ) for λc 0. Note that in [START_REF] Coster | Existence and multiplicity for elliptic p-Laplacian problems with critical growth in the gradient[END_REF] the above results were extended to the more general p-Laplacian case at the expense of considering µ constant. Also, in the frame of viscosity solutions and fully nonlinear equations, similar conclusions have been obtained in [START_REF] Nornberg | A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient[END_REF] under corresponding assumptions. All the above mentioned results require either µ to be constant or to be uniformly bounded from below by a positive constant (or similarly bounded from above by a negative constant). In [START_REF] Souplet | A priori estimates and bifurcation of solutions for an elliptic equation with semidefinite critical growth in the gradient[END_REF], assuming that λc, µ and h were non-negative, a first attempt to remove these restriction was presented. Under suitable assumptions on the support of the coefficient functions and for N ≤ 5, the existence of at least two solutions for λ > 0 small enough was obtained. Finally, let us point out that the only papers dealing with c which may change sign are [START_REF] Coster | A priori bounds and multiplicity of solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF][START_REF] Jeanjean | Multiple solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF]. In [START_REF] Jeanjean | Multiple solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF], the authors dealt with µ(x) ≡ µ > 0 constant and h 0 and they proved the existence of two solutions to (P ) for c + q and µh N /2 small enough. The restrictions µ > 0 constant and h 0 were removed in [START_REF] Coster | A priori bounds and multiplicity of solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF] at the expense of considering a "thick zero set" condition on the support of c and suitable assumptions on µ. Let us stress that [START_REF] Coster | A priori bounds and multiplicity of solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF] is the unique paper dealing with the non-coercive case c 0 where µ may change sign.

In this chapter we pursue the study of (P ) and consider several situations where c and h may change sign. At the expense of considering µ constant we remove the "thick zero set" condition on c considered in [START_REF] Coster | A priori bounds and multiplicity of solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF]. Moreover, we extend in several directions the previously known results and we clarify the structure of the set of solutions in the case where c + 0.

As a first main result, we completely characterize the limit coercive case. Let us consider the boundary value problem

-∆u = -d(x)u + µ|∇u| 2 + h(x), u ∈ H 1 0 (Ω) ∩ L ∞ (Ω), (3.1.1)
under the assumption

           Ω ⊂ R N , N ≥ 2, is a bounded domain with ∂Ω of class C 0,1 ,
d and h belong to L q (Ω) for some q > N /2, µ > 0 and d ≥ 0,

(3.1.2)
and define

m d :=          inf u∈W d Ω (|∇u| 2 -µh(x)u 2 )dx , if W d ∅ , + ∞ , if W d = ∅ , (3.1.3)
where

W d := {w ∈ H 1 0 (Ω) : d(x)w(x) = 0 a.e.
in Ω, w = 1}. We prove the following sharp result. As observed in [START_REF] Coster | A priori bounds and multiplicity of solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF], the structure of the set of solutions to (P ) depends on the size of c + but it is not affected by the size of c -. In order to clarify this, we replace c by a function c λ := λc +c - with λ a real parameter. More precisely, we consider the boundary value problem

-∆u = c λ (x)u + µ|∇u| 2 + h(x) , u ∈ H 1 0 (Ω) ∩ L ∞ (Ω) , (P λ )
under the assumption

           Ω ⊂ R N , N ≥ 2, is a bounded domain with ∂Ω of class C 0,1 ,
c + , c -and h belong to L q (Ω) for some q > N /2 , µ > 0 , c + 0 , c -≥ 0 and c + (x)c -(x) = 0 a.e. x ∈ Ω .

(A 1 )

Remark 3.1.2. Since h does not have a sign, there is no loss of generality in assuming µ > 0. If u is a solution to (P λ ) with µ < 0 then w = -u solves

-∆w = c λ (x)w -µ|∇w| 2 -h(x) , w ∈ H 1 0 (Ω) ∩ L ∞ (Ω).
Before going further and due to its importance on the rest of the chapter, let us stress that for λ = 0 the problem (P λ ) reduces to

-∆u = -c -(x)u + µ|∇u| 2 + h(x) , u ∈ H 1 0 (Ω) ∩ L ∞ (Ω). (P 0 )
Then, as an immediate corollary of Theorem 3.1.1, we have the following result.

Corollary 3.1.2. Assume that (A 1 ) holds. Then (P 0 ) has a solution if, and only if, m c -> 0.

Now, having at hand this satisfactory information about the limit coercive case, we turn to the study of the non-coercive case λ > 0. First, using mainly variational techniques, we prove the following theorem.

Theorem 3.1.3. Assume (A 1 ) and suppose that (P 0 ) has a solution. Then, there exists Λ > 0 such that, for all 0 < λ < Λ, (P λ ) has at least two solutions. Remark 3.1.3. This result improves and generalizes the main result obtained in [START_REF] Jeanjean | Multiple solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF].

Finally, we give sufficient conditions in terms of h ensuring that the hypotheses of Theorem 3.1.4 or of Theorem 3.1.5 are satisfied. Corollary 3.1.6. Assume that (A 2 ) holds and suppose that (P 0 ) has a solution:

• If h 0, then the conclusions of Theorem 3.1.4 hold.

• If h 0, then the conclusions of Theorem 3.1.5 hold. Remark 3.1.5. In case h 0, the problem (P 0 ) has always a solution.

We provide now some ideas of the proofs of Theorems 3.1.3, 3.1.4 and 3.1.5. First of all we should notice that, as µ is assumed to be a constant, we can perform a Hopf-Cole change of variable and reduce (P λ ) to a semilinear problem. Considering

v = 1 µ e µu -1 , (3.1.4)
one can check that u is a solution to (P λ ) if, and only if, v > -1/µ is a solution to

-∆v = c λ (x)g(v) + (1 + µv)h(x) , v ∈ H 1 0 (Ω) , (3.1.5)
where g is given by

g(s) = 1 µ (1 + µs) ln(1 + µs) , for s > -1/µ .
Hence, we need a control from below on the solutions to (3.1.5). More precisely, if v is a solution to (3.1.5), we need to verify that v > -1/µ. To this aim, in Section 3.4, we construct a lower solution u λ to (P λ ) below every upper solution to this problem. The fact that c λ has no sign causes several difficulties in this construction. We refer to Proposition 3.4.2 for more details. This lower solution allows us to introduce a new problem, which is completely equivalent to (P λ ). We define

α λ = 1 µ e µu λ -1
and introduce the problem

-∆v = f λ (x, v) , v ∈ H 1 0 (Ω) , (Q λ )
where

f λ (x, s) = c λ (x)g(s) + (1 + µs)h(x) , if s ≥ α λ (x) , c λ (x)g(α λ (x)) + (1 + µα λ (x))h(x) , if s ≤ α λ (x) . (3.1.6)
Then, we show that u is a solution to (P λ ) if, and only if, v defined by (3.1.4) is a solution to (Q λ ).

The main advantage of the problem (Q λ ) which respect to (P λ ) is that it admits a variational formulation. We shall then look for solutions to (Q λ ) as critical points of an associated functional

I λ : H 1 0 (Ω) → R defined as I λ (v) = 1 2 Ω |∇u| 2 dx - Ω F λ (x, v)dx , where G(s) = s 0 g(t)dt, F λ (x, s) = c λ (x) G(s) + 1 2µ (1 + µs) 2 h(x) , if s ≥ α λ (x) , Notation. 1) In R N , we use the notations |x| = x 2 1 + . . . + x 2 N and B R (y) = {x ∈ R N : |x -y| < R}.
2) We denote R + = (0, +∞) and R -= (-∞, 0).

3) For v ∈ L 1 (Ω) we define v + = max(v, 0) and v -= max(-v, 0).

4) For a, b ∈ L 1 (Ω) we denote {a ≤ b} = {x ∈ Ω : a(x) ≤ b(x)} .
5) The space H

* = +∞ in case N = 2. The norm in L ∞ (Ω) is u ∞ = esssup x∈Ω |u(x)|.

Preliminaries

This section presents some definitions and known results which are going to play an important role throughout the work. Let us start with some theory of lower and upper solution. We consider the boundary value problem

-∆u + H(x, u, ∇u) = ξ(x) , u ∈ H 1 0 (Ω) ∩ L ∞ (Ω) . (3.2.1)
where ξ belongs to L 1 (Ω) and

H : Ω × R × R N → R
(Ω) ∩ L ∞ (Ω) is a lower solution to (3.2.1) if α + ∈ H 1 0 (Ω) and, for all ϕ ∈ H 1 0 (Ω) ∩ L ∞ (Ω) with ϕ ≥ 0, if follows that Ω ∇α∇ϕ dx + Ω H(x, α, ∇α)ϕ dx ≤ Ω ξ(x)ϕ dx . Similarly, β ∈ H 1 0 (Ω) ∩ L ∞ (Ω) is an upper solution to (3.2.1) if β -∈ H 1 0 (Ω) and, for all ϕ ∈ H 1 0 (Ω) ∩ L ∞ (Ω) with ϕ ≥ 0, if follows that Ω ∇β∇ϕ dx + Ω H(x, β, ∇β)ϕ dx ≥ Ω ξ(x)ϕ dx .
R + → R + and a function k ∈ L 1 (Ω) such that |H(x, s, ξ)| ≤ b(|s|)[k(x) + |ξ| 2 ], a.e. x ∈ Ω, ∀(s, ξ) ∈ R × R N .
If there exist a lower solution α and an upper solution β of (3.2.1) with α ≤ β, then there exists a solution u of (3.2.1) with α ≤ u ≤ β. Moreover, there exists u min (resp. u max ) minimum (resp. maximum) solution to (3.2.1) with α ≤ u min ≤ u max ≤ β and such that, every solution u of (3.2.1) with α ≤ u ≤ β satisfies u min ≤ u ≤ u max . Definition 3.2.2. A lower solution α ∈ C 1 (Ω) is said to be strict if every solution u of (3.2.1) with u ≥ α satisfies u α. Similarly, an upper solution β ∈ C 1 (Ω) is said to be strict if every solution u of (3.2.1) such that u ≤ β satisfies u β.

Now, we consider the boundary value problem

-∆v = f (x, v) , v ∈ H 1 0 (Ω), (3.2.2)
being f : Ω × R → R an L p -Carathéodory function (see [START_REF] Coster | Two-point boundary value problems: lower and upper solutions[END_REF] for the definition of L p -Carathéodory function) for some p > N , such that

|f (x, s)| ≤ C|s| N +2 N -2 + d(x),
for some C > 0 and d ∈ L 2N N +2 (Ω). This problem can be handled variationally. We consider the associated functional J :

H 1 0 (Ω) → R defined by J(v) := 1 2 Ω |∇v| 2 dx - Ω F(x, v) dx , where ∂ ∂s F(x, s) = f (x, s),
and we recall the following results.

M := v ∈ H 1 0 (Ω) : α ≤ v ≤ β .
Then the infimum of J on M is achieved at some v, and such v is a solution to (3.2.2).

Corollary 3.2.3. Assume that α and β are strict lower and upper solutions to (3.2.2) belonging to C 1 (Ω) and satisfying α β and let M be defined as in Proposition 3.2.2. Then the minimizer v of J on M is a local minimizer of the functional J in the C 1 0 -topology. Furthermore, this minimizer is a solution to (3.2.2) with α v β.

Proof. First of all observe that Proposition 3.2.2 implies the existence of v ∈ H 1 0 (Ω) solution to (3.2.2), which minimizes J on M = {v ∈ H 1 0 (Ω) : α ≤ v ≤ β}. Moreover, as f is an L p -Carathéodory function for some p > N , the classical regularity results imply that v ∈ C 1 (Ω). Since the lower and the upper solutions are strict, it follows that α v β and so, there is a C 1 0 -neighbourhood of v in M. Hence, it follows that v minimizes locally J in the C 1 0 -topology.

Proposition 3.2.4. [54, Theorem 8] [29, Theorem 1] Assume that there exist h ∈ L p (Ω) for some p > N and σ ≤ 2 * (p-1) p -1 such that |f (x, s)| ≤ h(x)(1 + |s| σ ), a.e. x ∈ Ω , ∀ s ∈ R ,
and let v ∈ H 1 0 (Ω) be a local minimizer of J for the C 1 0 -topology. Then v ∈ C 1 0 (Ω) and it is a local minimizer of J in the H 1 0 -topology. Remark 3.2.1. If f is an L ∞ -Carathéodory function the result holds under the growth condition

|f (x, s)| ≤ C(1 + |s| σ ), a.e. x ∈ Ω , ∀ s ∈ R ,
for some C > 0 and α ≤ 2 * -1. In that case, we are exactly in the framework of [START_REF] Brezis | H 1 versus C 1 local minimizers[END_REF].

Finally, we recall some abstract results in order to find critical points of J other than local minima. Definition 3.2.3. Let (X, • ) be a real Banach space with dual space (X * , • * ) and let Φ : X → R be a C 1 functional. The functional Φ satisfies the Palais-Smale condition at level c ∈ R if, for any Palais-Smale sequence at level c ∈ R, i.e. for any sequence {x n } ⊂ X with

Φ(x n ) → c and Φ (x n ) * → 0 ,
there exists a subsequence {x n k } strongly convergent in X.

Theorem 3.2.5. [13, Theorem 2.1] Let (X, • ) be a real Banach space. Suppose that Φ : X → R is a C 1 functional. Take two points e 1 , e 2 ∈ X and define

Γ := {ϕ ∈ C([0, 1], X) : ϕ(0) = e 1 , ϕ(1) = e 2 } ,
and c := inf ϕ∈Γ max t∈[0,1] Φ(ϕ(t)) .
Assume that Φ satisfies the Palais-Smale condition at level c and that c > max{Φ(e 1 ), Φ(e 2 )} .

Then, there is a critical point of Φ at level c, i.e. there exists x 0 ∈ X such that Φ(x 0 ) = c and Φ (x 0 ) = 0.

Theorem 3.2.6. [65, Corollary 1.6] Let (X, • ) be a real Banach space and let Φ : X → R be a C 1 functional. Suppose that u 0 ∈ X is a local minimum, i.e. there exists ε > 0 such that

Φ(u 0 ) ≤ Φ(u), for u -u 0 ≤ ε ,
and assume that Φ satisfies the Palais-Smale condition at any level d ∈ R. Then: i) either there exists 0 < γ < ε such that inf{Φ(u) : uu 0 = γ} > Φ(u 0 ), ii) or, for each 0 < γ < ε, Φ has a local minimum at a point u γ with u γ -u 0 = γ and Φ(u γ ) = Φ(u 0 ).

Solving the limit coercive case

This section is devoted to the proof of Theorem 3.1.1. Let us first recall some of the notation introduced in Section 3.1. For a function d ∈ L q (Ω) for some q > N /2 we recall that

W d :={w ∈ H 1 0 (Ω) : d(x)w(x) = 0 a.e. in Ω, w = 1}
and

m d :=          inf u∈W d Ω (|∇u| 2 -µh(x)u 2 )dx , if W d ∅, + ∞ , if W d = ∅. (3.3.1)
Let us emphasize that W 0 = {w ∈ H 1 0 (Ω) : w = 1} and immediately observe that W d ⊆ W 0 . Remark 3.4.2. The constant µ > 0 can be replaced by a function µ ∈ L ∞ (Ω) with µ(x) ≥ µ 1 > 0 a.e. in Ω and the result still holds true.

Corollary 3.4.3. Under the assumption (A 2 ), for any λ > 0, there exists u λ ∈ C 1 (Ω) strict lower solution to (P λ ) such that, every β ∈ C 1 (Ω) upper solution to (P λ ) satisfies β u λ .

Proof. Let β ∈ C 1 (Ω) be an upper solution to (P λ ). By Proposition 3.4.2, there exists u λ ∈ H 1 (Ω) ∩ L ∞ (Ω) lower solution to (P λ ) such that min{0, β} ≥ u λ . Moreover, under the assumption (A 2 ), this lower solution belongs to C 1 (Ω). Now, we introduce w = βu λ ≥ 0 and we are going to show that w 0. First of all, by the construction of u λ , observe that

-∆w ≥ (λc + (x) -c -(x))w + µ |∇β| 2 -|∇u λ | 2 + h + (x) + 1 = (λc + (x) -c -(x)w + µ ∇β + ∇u λ , ∇w + h + (x) + 1, in Ω .
Equivalently, it follows that

-∆w -µ ∇β + ∇u λ , ∇w + c -(x)w ≥ λc + (x)w + h + (x) + 1 ≥ 1 , in Ω .
On the other hand, observe that w ≥ 0 on ∂Ω. Hence, by Theorem 3.1.7, it follows that w 0 and so that β u λ .

Now, following [46, Section 5] we introduce some auxiliary functions which, together with the lower solution found in Proposition 3.4.2, will let us introduce the desired equivalent problem to (P λ ). These functions will be essential throughout the rest work. We define In the following lemma we recall some properties of these functions.

g(s) =          1 µ (1 + µs) ln(1 + µs) , s > -1/µ , 0 , s ≤ -1/µ ,
Lemma 3.4.4.

i) The function g is continuous on R, satisfies g > 0 on R + and there exists D > 0 with -D ≤ g ≤ 0 on R -. Moreover, G ≥ 0 on R.

ii) For any δ > 0, there exists c = c(δ, µ) > 0 such that, for any s > 1 µ , g(s) ≤ c s 1+δ .

iii) lim s→+∞ g(s)/s = +∞ and lim s→+∞ G(s)/s 2 = +∞.

iv) For any s ∈ R, it follows that g(s)s ≥ 0.

Proof. See [46, Lemma 5.1] for i), ii) and iii). See [76, Lemma 7] for iv).

Next, we define the function

α λ = 1 µ (e µu λ -1) ∈ H 1 (Ω) ∩ L ∞ (Ω) , (3.4.7)
where

u λ ∈ H 1 (Ω) ∩ L ∞ (Ω)
is the lower solution to (P λ ) obtained in Proposition 3.4.2. Before going further, since u λ ≤ 0, observe that 0 ≥ α λ ≥ -1/µ + ε for some ε > 0. Having at hand α λ , for any λ ∈ R, we consider the auxiliary problem

-∆v = f λ (x, v) , v ∈ H 1 0 (Ω) , (Q λ )
where

f λ (x, s) = c λ (x)g(s) + (1 + µs)h(x) , if s ≥ α λ (x) , c λ (x)g(α λ (x)) + (1 + µα λ (x))h(x) , if s ≤ α λ (x) , (3.4.8)
with g defined in (3.4.6), and we are going to prove that (Q λ ) is completely equivalent to (P λ ).

Following [START_REF] Coster | Existence and multiplicity for elliptic p-Laplacian problems with critical growth in the gradient[END_REF]Lemma 5.2] one can obtain the next lemma.

Lemma 3.4.5. Assume that (A 1 ) holds. Then, it follows that:

i) Every solution to (Q λ ) belongs to L ∞ (Ω). ii) Every solution v to (Q λ ) satisfies v ≥ α λ . iii) A function v ∈ H 1 0 (Ω) is a solution to (Q λ ) if,

and only if, the function

u = 1 µ ln(1 + µv) ∈ H 1 0 (Ω) ∩ L ∞ (Ω)
is a solution to (P λ ).

Proof. See [46, Lemma

.

Remark 3.4.3. On the same way, we can show that

v 1 ∈ H 1 (Ω) ∩ L ∞ (Ω) (respectively v 2 ∈ H 1 (Ω) ∩ L ∞ (Ω)
) is a lower solution (respectively an upper solution) to (Q λ ) if, and only if, the function

u 1 = 1 µ ln 1 + µv 1 respectively u 2 = 1 µ ln 1 + µv 2
is a lower solution (respectively an upper solution) to (P λ ).

The problem (Q λ ) admits a variational formulation. Its solution in H 1 0 (Ω) can be obtained as critical points of the functional I λ : H 1 0 (Ω) → R defined as

I λ (v) = 1 2 Ω |∇v| 2 dx - Ω F λ (x, v)dx , (3.4.9)
where

F λ (x, s) = c λ (x) G(s) + 1 2µ (1 + µs) 2 h(x) , if s ≥ α λ (x) , (3.4.10)
and

F λ (x, s) = c λ (x)g(α λ (x)) + (1 + µα λ (x))h(x) (s -α λ (x)) + c λ (x) G(α λ (x)) + 1 2µ (1 + µα λ (x)) 2 h(x) , if s ≤ α λ (x) . (3.4.11)
Under the assumption (A 1 ), since g has subcritical growth (see Lemma 3.4.4), it is standard to show that I λ ∈ C 1 (H 1 0 (Ω), R).

On the Palais-Smale condition and the mountain pass geometry

Our first aim in this section is to show that, for any λ > 0, the functional I λ previously defined satisfies the Palais-Smale condition (Definition 3.2.3). Later on, we show that, if (P 0 ) has a solution, then the mountain pass geometry holds for λ > 0 small enough. As a consequence, we will prove Theorem 3.1.3.

Let us start showing that the Palais-Smale sequences are bounded. The proof is inspired by [START_REF] Jeanjean | Multiple solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF]. However, since we are not considering h 0, the proof becomes more difficult. The role of the lower solution obtained in Proposition 3.4.2 and the different definition of the functional are both crucial. Let us define

m c λ :=            inf u∈W c λ Ω (|∇u| 2 -µh(x)u 2 )dx , if W c λ ∅ , + ∞ , if W c λ = ∅ , (3.5.1)
where

W c λ = w ∈ H 1 0 (Ω) , c λ (x)w(x) = 0 a.e. x ∈ Ω , w ≥ 0 , w = 1 . Remark 3.5.1. Observe that W c λ ⊆ W c -and so that m c λ ≥ m c -.
Lemma 3.5.1. Fixed λ > 0 arbitrary, assume (A 1 ) and suppose that m c λ > 0. Then, the Palais-Smale sequences for I λ at any level d ∈ R are bounded.

Proof. Let {v n } ⊂ H 1 0 (Ω) be a Palais-Smale sequence for I λ at level d ∈ R. First we claim that {v - n } is bounded. Indeed, since {v n } is a Palais-Smale sequence, there exists a sequence ε n → 0 such that

-ε n v - n ≤ I λ (v n ), v - n ≤ ε n v - n . (3.5.2) Also, since f λ (x, s) is bounded in Ω × R -, there exist D 1 , D 2 > 0 such that I λ (v n ), v - n ≤ -v - n 2 + D 1 v - n + D 2 . ( 3
≤ -v - n 2 + (D 1 + ε n ) v - n + D 2 ,
and the claim follows. To prove that {v + n } is also bounded we assume by contradiction that v n → ∞ and introduce the sequence

{w n } ⊂ H 1 0 (Ω) given by w n = v n v n
. Observe that {w n } is bounded in

H 1 0 (Ω).
Hence, up to a subsequence, it follows that w n w in H 1 0 (Ω), w n → w strongly in L r (Ω) for 1 ≤ r < 2 * and w n → w a.e. in Ω. We split the rest of the proof into several steps:

Step 1: w ≡ 0.

As v - n is bounded and by assumption v n → ∞, clearly w -≡ 0. It then remains to prove that w + ≡ 0. We first prove that c λ w + ≡ 0. Assume by contradiction that c λ w + 0. Observe that for every ϕ ∈ H 1 0 (Ω), we can write

I λ (v n ), ϕ = Ω ∇v n ∇ϕ dx - {v n ≥α λ } µv n h(x)ϕ dx - {v n ≥α λ } c λ (x)g(v n )ϕ dx - {v n ≥α λ } h(x)ϕ dx - {v n ≤α λ } f λ (x, v n )ϕ dx (3.5.4)
Hence, using that f λ (x, s) is bounded in Ω × R -and the convergence of w n , it follows that

I λ (v n ), ϕ v n = Ω ∇w∇ϕ dx - {w≥α λ } µwh(x)ϕ dx - {v n ≥α λ } c λ (x)g(v n ) ϕ v n dx + o(1) , ∀ ϕ ∈ H 1 0 (Ω) .
Actually, using that g is bounded on R -and that w -≡ 0, we obtain that

Ω c λ (x)g(v n ) ϕ v n dx = Ω ∇w∇ϕ -µwh(x)ϕ dx + o(1) , ∀ ϕ ∈ H 1 0 (Ω) .
Equivalently, we deduce that, for every ϕ ∈ H 1 0 (Ω),

Ω c λ (x) g(v n ) -v n v n ϕ dx = Ω ∇w∇ϕ -(c λ (x) + µh(x))wϕ dx + o(1). ( 3.5.5)
Since c λ w + 0, we may choose ϕ ∈ H 1 0 (Ω) and a measurable subset

Ω ϕ ⊂ Ω such that |Ω ϕ | > 0 , c λ w + ϕ > 0 in Ω ϕ ⊂ Ω and c λ w + ϕ ≡ 0 in Ω \ Ω ϕ .
As g(s)s ≥ 0 on R (see Lemma 3.4.4), it follows that

c λ (x) g(v n ) -v n v n ϕ ≥ 0 a.e. in Ω ϕ .
Moreover, observe that

lim inf n→∞ c λ (x) g(v n ) -v n v n ϕ = lim inf n→∞ c λ (x)w n g(w n v n ) -w n v n w n v n ϕ = +∞ a.e. in Ω ϕ .
Hence, applying Fatou's Lemma we deduce that lim inf

n→∞ Ω ϕ c λ (x) g(v n ) -v n v n ϕdx = +∞ ,
which yields a contradiction with (3.5.5). Thus, we conclude that c λ w ≡ 0. Now, we take ϕ = w in (3.5.4) and divide by v n . Using that c λ w ≡ 0 and that {v n } is a Palais-Smale sequence, we get that

Ω ∇w n ∇wdx - Ω µw n wh(x)dx → 0,
and so, since w n w in H 1 0 (Ω) and

w n → w in L r (Ω), for 1 ≤ r < 2 * , that Ω |∇w| 2 -µh(x)w 2 dx = 0 .
By this last identity and the facts that w ≥ 0 and c λ w ≡ 0, the condition m c λ > 0 implies that w ≡ 0.

Step 2:

Ω c λ (x) g(v + n ) v + n (w + n ) 2 dx → 1.
First of all, observe that

I λ (v n ), v n v n 2 = 1 - 1 v n 2 µ {v n ≥α λ } (v 2 n )h(x)dx + {v n ≥α λ } c λ (x)g(v n )v n dx + {v n ≥α λ } v n h(x)dx + {v n ≤α λ } f λ (x, v n )dx → 0 .
Hence, using that w ≡ 0, we deduce that

1 - 1 v n 2 {v n ≥α λ } c λ (x)g(v n )v n dx → 0 .
Moreover, observe that

1 v n 2 {v n ≥α λ } c λ (x)g(v n )v n dx = 1 v n 2 Ω c λ (x)g(v + n )v + n dx + 1 v n 2 {α λ ≤v n ≤0} c λ (x)g(v n )v n dx ,
and 1 v n 2 {α λ ≤v n ≤0} c λ (x)g(v n )v n dx → 0 .
Thus, we can conclude that

1 - Ω c λ (x) g(v + n ) v + n (w + n ) 2 dx = 1 - 1 v n 2 Ω c λ (x)g(v + n )v + n dx → 0 .
Step 3:

ln( v n ) Ω c λ (x)(w + n ) 2 dx + Ω c λ (x)(w + n ) 2 ln µw + n + 1 v n dx → 1.
By the definition of g (see (3.4.6)), we can write

g(v + n ) v + n = 1 µv + n ln(1 + µv + n ) + ln(1 + µv + n ) = ln(1 + µ v n w + n ) µ v n w + n + ln(1 + µ v n w + n ) . Now, observe that c λ (x) ln(1 + µ v n w + n ) µ v n w + n ≤ |c λ (x)|,
and so, since w ≡ 0, that

1 µ Ω c λ (x) ln(1 + µ v n w + n ) µ v n w + n (w + n ) 2 dx → 0 .
Applying the previous step, we conclude that

Ω c λ (x)(w + n ) 2 ln(1 + µ v n w + n )dx → 1. (3.5.6) Observe that ln(1 + µ v n w + n ) = ln v n µw + n + 1 v n = ln( v n ) + ln µw + n + 1 v n .
Thus, substituting in (3.5.6), we conclude the Step 3.

Step 4:

ln( v n ) Ω c λ (x)(w + n ) 2 dx → 0.
First of all, defining

H(s) = 1 2 g(s)s -G(s),
observe that

I λ (v n ) - 1 2 I λ (v n ), v n = {v n ≥α λ } c λ (x)H(v n )dx - 1 2µ {v n ≥α λ } (1 + µv n )h(x)dx - {v n ≤α λ } F λ (x, v n ) - 1 2 f λ (x, v n )v n dx = Ω c λ (x)H(v + n )dx + {α λ ≤v n ≤0} c λ (x)H(v n )dx - 1 2µ {v n ≥α λ } (1 + µv n )h(x)dx - {v n ≤α λ } F λ (x, v n ) - 1 2 f λ (x, v n )v n dx = d + ε n v n + o(1) , or equivalently Ω c λ (x)H(v + n )dx = d - {α λ ≤v n ≤0} c λ (x)H(v n )dx + 1 2µ {v n ≥α λ } (1 + µv n )h(x)dx + {v n ≤α λ } F λ (x, v n ) - 1 2 f λ (x, v n )v n ) dx + ε n v n + o( 1) . (3.5.7)
Now, using that for every s ≥ 0,

H(s) = s 2 4 + s 2µ (1 -ln(1 + µs)) - 1 2µ 2 ln(1 + µs) ,
and substituting in (3.5.7), we deduce that

1 4 Ω c λ (x)(v + n ) 2 dx = d - {α λ ≤v n ≤0} c λ (x)H(v n )dx + 1 2µ {v n ≥α λ } (1 + µv n )h(x)dx + {v n ≤α λ } F λ (x, v n ) - 1 2 f λ (x, v n )v n dx - 1 2µ Ω c λ (x)v + n (1 -ln(1 + µv + n ))dx + 1 2µ 2 Ω c λ (x) ln(1 + µv + n )dx + ε n v n + o(1) .
As a consequence, we obtain that

ln( v n ) Ω c λ (x)(w + n ) 2 dx = 4 ln( v n ) v n 2       d - {α λ ≤v n ≤0} c λ (x)H(v n )dx + 1 2µ {v n ≥α λ } (1 + µv n )h(x)dx + {v n ≤α λ } F λ (x, v n ) - 1 2 f λ (x, v n )v n dx - 1 2µ Ω c λ (x)v + n (1 -ln(1 + µv + n ))dx + 1 2µ 2 Ω c λ (x) ln(1 + µv + n )dx + ε n v n       + o(1) .
We easily deduce that each term of the right hand side goes to zero. Thus, we can conclude that

ln( v n ) Ω c λ (x)(w + n ) 2 dx → 0 .
Step 5: Conclusion. Considering together Steps 3 and 4, we deduce that

Ω c λ (x)(w + n ) 2 ln µw + n + 1 v n → 1 ,
which clearly contradicts the fact that w ≡ 0. Since we have a contradiction, we conclude that v n is bounded, as desired.

Having at hand the boundedness of the Palais-Smale sequences, it is classical to show that the Palais-Smale condition holds. Proposition 3.5.2. Fixed λ > 0 arbitrary, assume that (A 1 ) holds and suppose that m c λ > 0. Then I λ satisfies the Palais-Smale condition at any level d ∈ R.

Proof. Thanks to Lemma 3.5.1 we know that the Palais-Smale sequences for I λ at any level d ∈ R are bounded. The strong convergence follows in a standard way. See [START_REF] Jeanjean | Existence and multiplicity for elliptic problems with quadratic growth in the gradient[END_REF]Lemma 11] or [START_REF] Coster | Existence and multiplicity for elliptic p-Laplacian problems with critical growth in the gradient[END_REF]Lemma 5.2] for two different approaches adapted to this setting. Now, we turn to check that the mountain pass geometry holds for λ > 0 small enough. We begin with a preliminary estimate. Lemma 3.5.3. Fixed Λ 1 > 0 arbitrary. There exist D 1 > 0 and D 2 > 0 such that, for every λ ∈ [0, Λ 1 ] and any v ∈ H 1 0 (Ω), it follows that

I λ (-v -) ≥ 1 2 v -2 -D 1 v --D 2 (3.5.8)
Proof. First of all, observe that for all v ∈ H 1 0 (Ω) we can write

I λ (-v -) = 1 2 v -2 - {0≥v≥α λ } (λc + (x) -c -(x))G(-v -) + 1 2µ (1 + 2µv) 2 h(x) dx - {α λ ≥v} (λc + (x) -c -(x))g(α λ ) + (1 + µα λ )h(x) (v -α λ )dx - {α λ ≥v} (λc + (x) -c -(x))G(α λ ) + 1 2µ (1 + µα λ ) 2 h(x) dx.
(3.5.9)

Hence, using that, for all λ ∈ R, α λ ∈ [-1/µ, 0] and Lemma 3.4.4, i), we have that, for all λ ∈ [0, Λ 1 ],

I λ (-v -) ≥ 1 2 v -2 - Ω λc + (x) max [-1/µ,0] G + 1 2µ h + (x) dx - {α λ ≥v} λc + (x)g(α λ ) -(1 + µα λ )h -(x) (v -α λ )dx ≥ 1 2 v -2 - Ω Λ 1 c + (x) max [-1/µ,0] G + 1 2µ h + (x) dx - Ω Λ 1 c + (x) max [-1/µ,0] |g| + h -(x) v -dx.
The estimate (3.5.8) follows immediately from the Sobolev inequality.

Lemma 3.5.4. Assume (A 1 ) and suppose that m c -> 0. Then, there exist constants Λ > 0 and R > 0 such that, if 0 ≤ λ ≤ Λ, then I λ (v) ≥ I λ (0) + 1 2 for all v ∈ ∂D with D := {v ∈ H 1 0 (Ω) : v + < R}. Proof. Let us begin with some preliminary observations. First of all, we fix Λ 1 > 0 arbitrary and, by Lemma 3.5.3, we know that, for all λ ∈ [0, Λ 1 ] and all v ∈ H 1 0 (Ω),

I λ (-v -) ≥ 1 2 v -2 -D 1 v --D 2 . (3.5.10)
This implies the existence of D 3 (independent of λ) such that, for all λ ∈ [0, Λ 1 ] and all v ∈ H 1 0 (Ω),

I λ (-v -) ≥ -D 3 . (3.5.11)
Now, by the definition of I λ and Lemma 3.4.4, observe that, for any λ ≥ 0 and any δ > 0, there exists D 4 > 0 (independent of λ) such that, for all v ∈ H 1 0 (Ω),

I λ (v + ) ≥ I 0 (v + ) -λD 4 (1 + v + 2+δ ). (3.5.12)
Also, since m c -> 0 by hypothesis, we know that I 0 is coercive (see [START_REF] Coster | Existence and multiplicity for elliptic p-Laplacian problems with critical growth in the gradient[END_REF]Proposition 6.1]) and so, that there exists R > 0 such that, for all v ∈ H 1 0 (Ω) with v + = R,

I 0 (v + ) ≥ - 1 µ Ω h(x)dx + 1 + D 3 . (3.5.13)
Gathering (3.5.12) and (3.5.13) we deduce the existence of 0 < Λ ≤ Λ 1 such that, for all λ ∈ [0, Λ] and all v ∈ H 1 0 (Ω) with v + = R, the following inequality holds

I λ (v + ) ≥ - 1 µ Ω h(x)dx + 1 2 + D 3 . (3.5.14)
Now, for the constants Λ > 0 and R > 0 previously given, we define D := {v ∈ H 1 0 (Ω) : v + < R} and consider an arbitrary λ ∈ [0, Λ]. In order to finish the proof, we are going to show that

I λ (v) ≥ I λ (0) + 1 2 , ∀ v ∈ ∂D.
Let v ∈ ∂D fixed but arbitrary. By (3.5.11), (3.5.14) and the fact that I λ (0) = -1 2µ Ω h(x)dx, we directly obtain that

I λ (v) = I λ (v + ) + I λ (-v -) -I λ (0) ≥ - 1 µ Ω h(x)dx + 1 2 + D 3 -D 3 + 1 2µ Ω h(x)dx = I λ (0) + 1 2 .
Lemma 3.5.5. Assume (A 1 ). For any λ > 0, M > 0 and R > 0, there exists w ∈ H 1 0 (Ω) such that w + > R and I λ (w) ≤ -M.

Proof. Since c + 0, we can choose v ∈ C ∞ 0 (Ω) such that v ≥ 0, c + v 0 and c -v ≡ 0. Moreover, let us take t ∈ R + , t ≥ 1. As α λ ≤ 0, observe that I λ (tv) ≤ 1 2 t 2 Ω |∇v| 2 -µh(x)v 2 dx -λt 2 Ω c + (x)v 2 G(tv) t 2 v 2 dx + t µ 1 + µv ∞ h - 1 = t 2 Ω |∇v| 2 -µh(x)v 2 dx -λ Ω c + (x)v 2 G(tv) t 2 v 2 dx + 1 t 1 µ 1 + µv ∞ h - 1 .
Now, since by Lemma 3.4.4, we have

lim t→∞ λ Ω c + (x)v 2 G(tv) t 2 v 2 dx = +∞ ,
we deduce that lim t→∞ I λ (tv) = -∞ and the lemma follows.

Gathering Lemmas 3.5.4 and 3.5.5 we deduce that, for λ > 0 small enough, I λ possess a mountainpass geometry. Once this is proved, we first show the existence of a local minimum of I λ and then we prove Theorem 3.1.3. Proposition 3.5.6. Assume (A 1 ) and suppose that m c -> 0 and that λ ≥ 0 is small enough in order to ensure that the conclusion of Lemma 3.5.4 holds. Then, I λ possesses a critical point v with I λ (v) ≤ I λ (0), which is a local minimum.

Proof. From Lemma 3.5.4, we know that there exist R > 0 such that

m := inf v∈D I λ (v) ≤ I λ (0)
and

I λ (v) > I λ (0) if v ∈ ∂D,
where

D := {v ∈ H 1 0 (Ω) : v + < R}. Let {v n } ⊂ D be such that I λ (v n ) → m.
By the definition of D and (3.5.10), we deduce that {v n } is bounded and so, up to a subsequence, it follows that v n v ∈ H 1 0 (Ω). By the weak lower semicontinuity of the norm and of the functional I λ , we have

v + ≤ lim inf n→∞ v + n ≤ R and I λ (v) ≤ lim inf n→∞ I λ (v n ) = m ≤ I λ (0) .
Finally, since, by Lemma 3.5.4, we know that

I λ (v) > I λ (0) if v ∈ ∂D, we deduce that v ∈ D is a local minimum of I λ .
Proof of Theorem 3.1.3. Assume that λ > 0 is small enough in order to ensure that the conclusion of Lemma 3.5.4 holds. By Proposition 3.5.6 we have a first critical point, which is a local minimum of I λ . On the other hand, since the Palais-Smale condition holds, in view of Lemmas 3.5.4. and 3.5.5, we can apply Theorem 3.2.5 and obtain a second critical point of I λ at the mountain-pass level. This gives two different solutions to (Q λ ). Finally, by Lemma 3.4.5, we obtain two different solutions to (P λ ).

3.6 Proof of Theorems 3.1.4 and 3.1.5

This section is devoted to the proofs of Theorems 3.1.4 and 3.1.5 and Corollary 3.1.6. Let us recall that, under the assumption (A 2 ), every solution to (P λ ) belongs to C 1 0 (Ω) (see [50, Theorem 2.2]). We first prove Theorem 3.1.4 and the first part of Corollary 3.1.6.

Lemma 3.6.1. [17, Lemma 2.2] Assume (A 1 ). If u 1 , u 2 ∈ H 1 (Ω) ∩ W 1,N loc (Ω) ∩ C(Ω)
are respectively a lower and an upper solution to (P 0 ), then u 1 ≤ u 2 . Lemma 3.6.2. Assume (A 2 ) and suppose that (P 0 ) has a solution u 0 with c + u 0 0. Then, it follows that:

• Every u solution to (P λ ) with λ > 0 and c + u ≥ 0 satisfies u u 0 .

• For all λ > 0, u 0 is a strict lower solution to (P λ ).

• There exists λ ∈]0, +∞[ such that, for all 0 < λ < λ, (P λ ) has a strict upper solution u 1 with u 1 u 0 and, for all λ > λ, (P λ ) has no solutions u with c + u ≥ 0.

Proof. Let us split the proof into four steps:

Step 1: Every u solution to (P λ ) with λ > 0 and c + u ≥ 0 satisfies u u 0 .

Let u be a solution to (P λ ) with λ > 0 and c + u ≥ 0. We easily observe that u ∈ C 1 0 (Ω) is an upper solution to (P 0 ) and so, applying Lemma 3.6.1, that u ≥ u 0 in Ω . Now, arguing as in Corollary 3.4.3, we are going to show that u u 0 . Let us define w = uu 0 and observe that

-∆w -µ ∇u + ∇u 0 , ∇w + c -(x)w ≥ λc + (x)u 0 ,
in Ω , w = 0 , on ∂Ω .

Applying then Theorem 3.1.7, we deduce that w 0 and so that u u 0 .

Step 2: For all λ > 0, u 0 is a strict lower solution to (P λ ).

As c + u 0 0, we easily observe that u 0 is a lower solution to (P λ ). Arguing as in Step 1, we deduce that u 0 is a strict lower solution to (P λ ).

Step 3: The problem (P λ ) has no solution u with c + u ≥ 0 for λ > 0 large.

We argue by contradiction. Suppose that u is a solution to (P λ ) with c + u ≥ 0 and let γ 1 > 0 be the first eigenvalue and ϕ 1 > 0 be the first eigenfunction to the eigenvalue problem

-∆v + c -(x)v = γc + (x)v , v ∈ H 1 0 (Ω) .
Multiplying (P λ ) by ϕ 1 and integrating it follows that

γ 1 Ω c + (x)uϕ 1 dx = Ω ∇u∇ϕ 1 + c -(x)uϕ 1 dx = λ Ω c + (x)uϕ 1 dx + µ Ω |∇u| 2 ϕ 1 dx + Ω h(x)ϕ 1 dx , or equivalently 0 = (λ -γ 1 ) Ω c + (x)uϕ 1 dx + µ Ω |∇u| 2 ϕ 1 dx + Ω h(x)ϕ 1 dx . Hence, as u ≥ u 0 , if λ > γ 1 it follows that 0 ≥ (λ -γ 1 ) Ω c + (x)u 0 ϕ 1 dx + Ω h(x)ϕ 1 dx . (3.6.1)
Since c + u 0 0 and ϕ 1 > 0 , (3.6.1) gives a contradiction for λ large enough.

Step 4: Let us define λ := sup {λ : (P λ ) has a solution u with c + u ≥ 0} . For all 0 < λ < λ , (P λ ) has a strict upper solution u 1 u 0 and, for all λ > λ, (P λ ) has no solutions with c + u ≥ 0.

First of all, observe that Step 2 implies λ < ∞. Furthermore, by the definition of λ, it is obvious that, for λ > λ, (P λ ) has no solutions u with c + u ≥ 0.

Let us then consider 0 < λ < λ. By the definition of λ, we can find λ ∈]λ, λ[ and u λ solution to (P λ ) with c + u λ ≥ 0. Then, observe that u λ ∈ C 1 0 (Ω) is an upper solution to (P 0 ) and, by Step 1, u λ u 0 .

Finally, we are going to show that u λ is a strict upper solution to (P λ ). Using that c + u λ ≥ 0 and λ > λ, we deduce that u λ is an upper solution to (P λ ). Finally, let us consider u a solution to (P λ ) with u ≤ u λ and introduce w = u λu. Arguing as in Corollary 3.4.3, we are going to show that w 0. Directly observe that

     -∆w -µ ∇u + ∇u λ , ∇w + c -(x)w ≥ ( λ -λ)c + (x)u 0 ,
in Ω , w = 0 , on ∂Ω .

Hence, applying Theorem 3.1.7, we obtain that w 0, and so that u λ is strict.

Proposition 3.6.3. Assume (A 2 ), suppose that (P 0 ) has a solution u 0 with c + u 0 0 and let λ ∈]0, +∞[ be given by Lemma 3.6.2. Then:

1) For every 0 < λ < λ, there exists v ∈ C 1 0 (Ω) with v v 0 := 1 µ (e µu 0 -1), which is a local minimum of I λ in the H 1 0 -topology and a solution to (Q λ ).

2) For λ = λ, there exists v ∈ C 1 0 (Ω) with v ≥ v 0 , which is a solution to (Q λ ).

Proof. 1) By Lemma 3.6.2, for all 0 < λ < λ, u 0 is a strict lower solution to (P λ ) and there exists u 2) Let {λ n } be a sequence with 0 < λ n < λ and λ n → λ and let {v n } be the corresponding sequence of minimum of I λ n obtained in 1). This implies that I λ n (v n ), ϕ = 0 for all ϕ ∈ H 1 0 (Ω). Moreover, by the construction of the lower and upper solutions, arguing as in Lemma 3.5.3, we obtain that

I λ n (v n ) ≤ I λ n (v 0 ) ≤ I λ (v 0 ) + D λ
for some D λ > 0. Then, arguing as in Lemma 3.5.1 and Proposition 3.5.2, we prove the existence of v ∈ H 1 0 (Ω) such that v n → v in H 1 0 (Ω) with v a solution to (Q λ ) for λ = λ. Moreover, as v n ≥ v 0 for all n ∈ N, we deduce that v ≥ v 0 .

Proof of Theorem 3.1.4. Let us define as in Lemma 3.6.2 and Proposition 3.6.3 λ := sup {λ : (P λ ) has a solution u with c + u ≥ 0} .

We split the proof into several steps:

Step 1: Every u solution to (P λ ) with c + u ≥ 0 satisfies u u 0 . This follows directly from Lemma 3.6.2.

Appendix. Hopf's Lemma ans SMP with unbounded lower order terms

In this section we prove Theorem 3.1.7 which can be seen as a combination of the Strong maximum principle and the Hopf's Lemma. The proof of Theorem 3.1.7 will be obtained as a consequence of the Hopf's Lemma with unbounded lower order term that we prove in Lemma 3.7.6. We give here a simplified proof of [START_REF] Rosales | Generalizing Hopf's boundary point Lemma[END_REF]Theorem 4.1]. Let us begin with some preliminary results that will be needed to prove Lemma 3.7.6. Throughout the appendix we assume N ≥ 2. As a consequence of the previous lemma we obtain an existence result with inhomogeneous boundary conditions. Corollary 3.7.2. Let ω = B 1 (0) \ B 1/2 (0), β ∈ (L p (ω)) N and ξ ∈ L p (ω) for some p > N and assume that ξ ≥ 0 a.e. in ω. Then, there exists an unique u ∈ C 1,τ (ω) for some τ > 0 solution to

           -∆u + β(x), ∇u + ξ(x)u = 0, in ω, u = 0, on ∂B 1 (0), u = 1, on ∂B 1/2 (0), (3.7.2
)

such that 0 ≤ u ≤ 1 in ω.
Proof. Let us consider ϕ ∈ C ∞ (R N ) given by ϕ(x) = 4 3 (1 -|x| 2 ), and observe that ϕ(x) = 0 for all x ∈ ∂B 1 (0) and ϕ(x) = 1 for all x ∈ ∂B 1/2 (0). Moreover, by direct computations it follows that

-∆ϕ + β(x), ∇ϕ + ξ(x)ϕ = 8N 3 - 8 3 β(x), x + 4 3 ξ(x)(1 -|x| 2 ) =: -F ∈ H -1 (ω).
By Lemma 3.7.1 we know that there exists an unique w ∈ H 1 0 (ω) solution to Proof. First of all, observe that

-∆w + β(x), ∇w + ξ(x)w = F, in ω, w = 0, on ∂ω. ( 3
T (ω) = B 1/ε 1 - 1 ε x 0 \ B 1/2ε 1 - 1 ε x 0 = x ∈ R N : 1 2ε < x -1 - 1 ε x 0 < 1 ε . (3.7.4)
Now, observe that, for all x ∈ ω and all ε ∈ (0, 1/4), it follows that

x -1 - 1 ε x 0 ≤ |x| + 1 - 1 ε |x 0 | = |x| + 1 ε -1 < 1 + 1 ε -1 = 1 ε , (3.7.5) and x -1 - 1 ε x 0 ≥ |x| -1 - 1 ε |x 0 | = 1 ε -1 -|x| > 1 ε -2 > 1 2ε . ( 3
• B i ε L p (ω) ≤ ε 1-N p B i L p (ω) for all i = 1, . . . , N . • c ε L p (ω) ≤ ε 2-N p c L p (ω)
Proof. Let i ∈ {1, . . . , N }. We directly observe that

B i ε p L p (ω) = ω |B i ε (y)| p dy = ε p ω |B i (ε(y -x 0 ) + x 0 )| p dy = ε p-N S(ω) |B i (z)| p dz, ( 3.7.7)
where z = S(y) = ε(yx 0 ) + x 0 . Then, arguing as in Lemma 3.7.3, we obtain that S(ω) ⊂ ω, and so, taking into account (3.7.7), we deduce that

B i ε p L p (ω) ≤ ε p-N ω |B i (z)| p dz = ε p-N B i p L p (ω) .
The estimate for c ε follows arguing on the same way.

Using the rescaled functions B ε and c ε defined in Lemma 3.7.4, we introduce the auxiliary boundary value problem

           -∆u + B ε (x), ∇u + c ε (x)u = 0, in B 1 (0) \ B 1/2 (0), u = 0, on ∂B 1 (0), u = 1, on ∂B 1/2 (0). (P ε )
and we prove the following uniform a priori bound that will be crucial in the proof of Lemma 3.7.6. Proof. We argue by contradiction. We assume that there exist sequences {ε n } ⊂ [0, 1/4] and {u n } solutions to (P ε ) with ε = ε n such that

u n C 1 (ω) → +∞, as n → ∞.
Without loss of generality (up to a subsequence if necessary) we may assume that

1 ≤ u n C 1 (ω) , ∀ n ∈ N.
Step 1: Auxiliary regular barrier ϕ

Let us consider the problem

           -∆ϕ = 0, in ω, ϕ = 0, on ∂B 1 (0), ϕ = 1,
on ∂B 1/2 (0).

(3.7.12)

By [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Theorem 6.14] we know that there exists ϕ ∈ C 2,τ (ω) for some τ > 0 solution to (3.7.12). Moreover, by [66, Lemma 3.4 and Theorem 3.5], we know that 0 < ϕ(x) < 1, ∀ x ∈ ω, and ∂ϕ ∂ν (x 0 ) < 0.

(3.7.13)

Step 2: Let M > 0 given by Lemma 3.7.5. For every ε ∈ (0, 1/4) there exists ϕ ε ∈ C 1,τ (ω) for some τ > 0 solution to

(P ε ) such that ϕ ε C 1 (ω) ≤ M.
The existence follows from Corollary 3.7.2 and the uniform bound from Lemma 3.7.5.

Step 3: Let ϕ ε the solution to (P ε ) given by Step 2. There exists ε ∈ (0, 1/4) such that, for all ε ∈ (0, ε), it follows that ∂ϕ ε ∂ν (x 0 ) < 0. Let us define ψ ε := ϕ εϕ and observe that ψ ε ∈ C 1,τ (ω) for some τ > 0 and solves

-∆ψ ε = -B ε (x), ∇ϕ ε -c ε (x)ϕ ε , in ω, ψ ε = 0, on ∂ω. ( 3.7.14)
Then, by [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Lemma 9.17] and Lemma 3.7.4, there exists C = C(ω, N ) > 0 such that

ψ ε W 2,p (ω) ≤ C B ε (x), ∇ϕ ε + c ε (x)ϕ ε L p (ω) ≤ C ϕ ε C 1 (ω) ε 1-N p        N i=1 B i L p (ω) + ε c L p (ω)        ≤ ε 1-N p CM        N i=1 B i L p (ω) + c L p (ω)        =: ε 1-N p C 2 (3.7.15)
for some C 2 independent of ε. Hence, by the Sobolev continuous embedding, there exists C 3 > 0 independent of ε such that

ψ ε C 1,τ (ω) ≤ ε 1-N p C 3 .
We conclude that

lim ε→0 ∂ϕ ε ∂ν (x 0 ) - ∂ϕ ∂ν (x 0 ) ≤ lim ε→0 ψ ε C 1,τ (ω) = 0,
and the Step 3 follows.

Step 4: Conclusion

Let u ∈ C 1 (B 1 (0)) be an upper solution to (3.7.11) such that u(x) > u(x 0 ) = 0 for all x ∈ B 1 (0). We fix ε > 0 small enough to ensure that the Step 3 holds and define u ε (y) = u(ε(yx 0 ) + x 0 ). Since we know that ω ⊂ T (ω) by Lemma 3.7.3, we have that u ε is an upper solution to

-∆u ε + B ε (x), ∇u ε + c ε (x)u ε = 0, in ω, u ε = 0, on ∂ω. (3.7.16)
Then, we define u ε = u εθ ε ϕ ε with

θ ε = inf ∂B 1/2 (0) u ε > 0,
and we have that u ε is an upper solution to

-∆u ε + B ε (x), ∇u ε + c ε (x)u ε = 0, in ω, u ε = 0, on ∂ω.
(3.7.17)

Applying then [START_REF] Kim | Green's function for second order elliptic equations with singular lower order coefficients[END_REF]Proposition 3.10] we deduce that u εθ ε ϕ ε ≥ 0, in ω, and so, by Step 3, we conclude that ∂u ∂ν

(x 0 ) = 1 ε ∂u ε ∂ν (x 0 ) ≤ θ ε ε ∂ϕ ε ∂ν (x 0 ) < 0,
as desired. Observe that if u is an upper solution to (3.7.18) such that u(x) > 0 for all x ∈ B R (z), then v is an upper solution to

-∆v + B(y), ∇v + c(y)v = 0, in B 1 (0), v = 0, on ∂B 1 (0), (3.7.20)
satisfying the hypothesis of Lemma 3.7.6 for some y 0 = T (x 0 ) ∈ ∂B 1 (0). Thus, we have that

∂u ∂ν (x 0 ) = 1 R ∂v ∂ν (y 0 ) < 0,
and the result follows.

Proof of Theorem 3.1.7. The result follows from Corollary 3.7.7 arguing as in [START_REF] Tolksdorf | On the Dirichlet problem for quasilinear equations in domains with conical boundary points[END_REF]Theorem 3.27].

See also [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Theorem 3.5].

4

Nonlinear fractional Laplacian problems with nonlocal "gradient terms"

Introduction and main results

In the last fifteen years, there has been an increasing interest in the study of partial differential equations involving integro-differential operators. In particular, the case of the fractional Laplacian has been widely studied and is nowadays a very active field of research. This is due not only to its mathematical richness. The fractional Laplacian has appeared in a great number of equations modeling real world phenomena, especially those which take into account nonlocal effects. Among others, let us mention applications in quasi-geostrophic flows [START_REF] Caffarelli | Drift diffusion equations with fractional diffusion and the quasigeostrophic equation[END_REF], quantum mechanic [START_REF] Laskin | Fractional quantum mechanics and lévy path integrals[END_REF], mathematical finances [START_REF] Applebaum | Lévy processes and stochastic calculus[END_REF][START_REF] Cont | Financial modelling with jump processes[END_REF], obstacle problems [START_REF] Barrios | Free boundary regularity in the parabolic fractional obstacle problem[END_REF][START_REF]Global regularity for the free boundary in the obstacle problem for the fractional Laplacian[END_REF][START_REF] Caffarelli | Regularity of solutions to the parabolic fractional obstacle problem[END_REF] and crystal dislocation [START_REF] Dipierro | Strongly nonlocal dislocation dynamics in crystals[END_REF][START_REF] Dipierro | Dislocation dynamics in crystals: a macroscopic theory in a fractional Laplace setting[END_REF][START_REF] Toland | The Peierls-Nabarro and Benjamin-Ono equations[END_REF].

The first aim of the present chapter is to discuss, depending on the real parameter λ > 0, the existence and non-existence of solutions to the Dirichlet problem

       (-∆) s u = µ(x) D 2 s (u) + λf (x) , in Ω, u = 0 , in R N \ Ω, (P λ )
under the assumption

           Ω ⊂ R N , N ≥ 2, is a bounded domain with ∂Ω of class C 2 , s ∈ (1/2, 1), f ∈ L m (Ω)
for some m > N /2s and µ ∈ L ∞ (Ω).

(A 1 )

Throughout the chapter, (-∆) s stands for the, by know classical, fractional Laplacian operator. For a smooth function u and s ∈ (0, 1), it can be defined as

(-∆) s u(x) := a N ,s p.v. R N u(x) -u(y) |x -y| N +2s dy, where a N ,s := R N 1 -cos(ξ 1 ) |ξ| N +2s dξ -1 = - 2 2s Γ N 2 + s π N 2 Γ (-s)
, is a normalization constant and "p.v." is an abbreviation for "in the principal value sense". In (P λ ), D 2 s is a nonlocal diffusion term. It plays the role of the "gradient square" in the nonlocal case and is given by

D 2 s (u) = a N ,s 2 p.v. R N |u(x) -u(y)| 2 |x -y| N +2s dy . (4.1.1)
Since they will not play a role in this thesis, we normalize the constants appearing in the definitions of (-∆) s and D 2 s and we omit the p.v. sense. However, let us stress that these constants guarantee

lim s→1 -(-∆) s u(x) = -∆u(x), ∀ u ∈ C ∞ 0 (R N ), (4.1.2) and lim s→1 -D 2 s (u(x)) = |∇u(x)| 2 , ∀ u ∈ C ∞ 0 (R N ). (4.1.3)
We refer to [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF] and [START_REF] Caffarelli | Interior regularity for fractional systems[END_REF] respectively for a proof of (4.1.2) and (4.1.3). Hence, at least formally, if s → 1 -in (P λ ), we recover the local problem

-∆u = µ(x)|∇u| 2 + λf (x), in Ω, u = 0, on ∂Ω. ( 4.1.4)
This equation corresponds to the stationary case of the Kardar-Parisi-Zhang model of growing interfaces introduced in [START_REF] Kardar | Dynamic scaling of growing interfaces[END_REF]. The existence and multiplicity of solutions to problem (4.1.4) and of its different extensions have been extensively studied and it is still an active field of research. See for instance [4,[START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF][START_REF]L ∞ estimate for some nonlinear elliptic partial differential equations and application to an existence result[END_REF][START_REF] Coster | Existence and multiplicity for elliptic p-Laplacian problems with critical growth in the gradient[END_REF][START_REF] Ferone | Nonlinear problems having natural growth in the gradient: an existence result when the source terms are small[END_REF][START_REF] Grenon | Existence and a priori estimate for elliptic problems with subquadratic gradient dependent terms[END_REF]. In most of these papers, the existence of solutions is proved using either a priori estimates or, when it is possible, a suitable change of variable to obtain an equivalent semilinear problem. However, neither of these techniques seem to be appropriate to deal with the nonlocal case (P λ ).

Let us also point out that in [START_REF]Semilinear fractional elliptic equations with gradient nonlinearity involving measures[END_REF], using pointwise estimates on the Green function for the fractional Laplacian, the authors deal with the nonlocal-local problem

       (-∆) s u = |∇u| q + λf (x), in Ω, u = 0, in R N \ Ω. (4.1.5)
For s ∈ (1/2, 1), 1 < q < N N -(2s-1) , f ∈ L 1 (Ω) and λ > 0 small enough they obtained the existence of a solution to (4.1.5). This existence result was later completed in [START_REF]Towards a deterministic KPZ equation with fractional diffusion: the stationary problem[END_REF] where, under suitable assumptions on f , the authors showed the existence of a solution to (4.1.5) for all 1 < q < ∞ and λ > 0 small enough.

Following [START_REF] Chen | Semilinear fractional elliptic equations involving measures[END_REF][START_REF]Semilinear fractional elliptic equations with gradient nonlinearity involving measures[END_REF] we introduce the following notion of weak solution to (P λ ). In the spirit of the existing results for the local case, our first main result shows the existence of a weak solution to (P λ ) under a smallness condition on λf . Theorem 4.1.1. Assume that (A 1 ) holds. Then there exists λ * > 0 such that, for all 0 < λ ≤ λ * , (P λ ) has a weak solution u ∈ W s,2 0 (Ω) ∩ C 0,α (Ω) for some α > 0.

Remark 4.1.1.

a) The definition of W s,2 0 (Ω) will be introduced in Section 4.2.

b) In 1983, L. Boccardo, F. Murat and J.P. Puel [START_REF] Boccardo | Existence de solutions faibles pour des équations elliptiques quasilinéaires à croissance quadratique[END_REF] already pointed out that the existence of solution to (4.1.4) is not guaranteed for every λf ∈ L ∞ (Ω). Some extra conditions are needed. Hence, the smallness condition appearing in Theorem 4.1.1 was somehow expected. c) For λf ≡ 0, u ≡ 0 is a solution to (P λ ) that obviously belongs to W s,2 0 (Ω) ∩ C 0,α (Ω). Hence, there is no loss of generality to assume that λ > 0.

The counterpart of |∇u| 2 in (4.1.4) is played in (P λ ) by D 2 s (u). This term appears in several applications. For instance, let us mention [START_REF] Caffarelli | Interior regularity for fractional systems[END_REF][START_REF] Millot | On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into spheres[END_REF][START_REF] Schikorra | Integro-differential harmonic maps into spheres[END_REF] where it naturally appears as the equivalent of |∇u| 2 when considering fractional harmonic maps into the sphere.

Let us now give some ideas of the proof of Theorem 4.1.1. As in the local case, see for instance [START_REF] Phuc | Morrey global bounds and quasilinear Riccati type equations below the natural exponent[END_REF], the existence of solutions to (P λ ) is related to the regularity of the solutions to a linear equation of the form Having at hand suitable regularity results for (4.1.8) and inspired by [94, Section 6], we develop a fixed point argument to obtain a solution to (P λ ). Note that, due to the nonlocality of the operator and of the "gradient term", the approach of [START_REF] Phuc | Morrey global bounds and quasilinear Riccati type equations below the natural exponent[END_REF] has to be adapted significantly. In particular, the form of the set where we apply the fixed point argument seems to be new in the literature. We consider a subset of W s,1 0 (Ω) where, in some sense, we require more "differentiability" and more integrability. This extra "differentiability" is a purely nonlocal phenomena and it is related with our regularity results for (4.1.8). See Section 4.4 for more details.

       (-∆) s v = h(x), in Ω, v = 0, in R N \ Ω. ( 4
Let us also stress that the restriction s ∈ (1/2, 1) comes from the regularity results of Section 4.3. If suitable regularity results for (4.1.8) with s ∈ (0, 1/2] were available, our fixed point argument would provide the desired existence results to (P λ ). See Section 4.3 for more details.

Next, let us prove that the smallness condition imposed in Theorem 4.1.1 is somehow necessary.

Theorem 4.1.2. Assume (A 1 ) and suppose that µ(x) ≥ µ 1 > 0 and f + 0. Then there exists λ * * > 0 such that, for all λ > λ * * , (P λ ) has no weak solutions in W s,2 0 (Ω). Remark 4.1.2. a) Observe that, if v is a solution to

       (-∆) s v = -µ(x) D 2 s (v) -λf (x) , in Ω, v = 0 , in R N \ Ω,
then u = -v is a solution to (P λ ). Hence, if µ(x) ≤ -µ 1 < 0 and f -0 we recover the same kind of non-existence result and the smallness condition is also required.

b) Since we do not use the regularity results of Section 4.3, the restriction s ∈ (1/2, 1) is not necessary in the proof of Theorem 4.1.2. The result holds for all s ∈ (0, 1).

Also, in order to show that the regularity imposed on the data f is almost optimal, we provide a counterexample to our existence result when the regularity condition on f is not satisfied. The proof makes use of the Hardy potential.

       (-∆) s u = µ(x) u D 2 s (u) + λf (x) , in Ω, u = 0 , in R N \ Ω. ( P λ )
For µ(x) ≡ 1, this problem can be seen as a particular case of the fractional harmonic maps problem considered in [START_REF] Caffarelli | Interior regularity for fractional systems[END_REF][START_REF] Millot | On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into spheres[END_REF].

Remark 4.1.3. The notion of weak solution to ( P λ ) is essentially the same as in Definition 4.1.1. The only difference is that we now require that u and u D 2 s (u) belong to L 1 (Ω). We derive the following existence result for λf small enough. Theorem 4.1.4. Assume that (A 1 ) holds. Then, there exists λ * > 0 such that, for all 0 < λ ≤ λ * , ( P λ ) has a weak solution u ∈ W s,2 0 (Ω) ∩ C 0,α (Ω) for some α > 0. Next, motivated by some other results on fractional harmonic maps into the sphere [START_REF] Da Lio | Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps[END_REF][START_REF] Da Lio | n/p-harmonic maps: regularity for the sphere case[END_REF] and some classical results of harmonic analysis [109, Chapter V], we consider a different diffusion term. Depending on the real parameter λ > 0, we study the existence of solutions to the Dirichlet problem

       (-∆) s u = µ(x)|(-∆) s 2 u| q + λf (x) , in Ω, u = 0 , in R N \ Ω, (Q λ )
under the assumption

             Ω ⊂ R N , N ≥ 2, is a bounded domain with ∂Ω of class C 2 , f ∈ L m (Ω) for some m ≥ 1 and µ ∈ L ∞ (Ω),
s ∈ (1/2, 1) and 1 < q < N Nms .

(B 1 ) Remark 4.1.4. If m ≥ N /s, we just need to assume 1 < q < ∞ in (B 1 ).

Since the diffusion term considered in (Q λ ) is different from the ones in (P λ ) and ( P λ ), we shall make precise the notion of weak solution to (Q λ ). Definition 4.1.2. We say that u is a weak solution to

(Q λ ) if u ∈ L 1 (Ω), |(-∆) s 2 u| ∈ L q (Ω), u ≡ 0 in CΩ and Ω u(-∆) s φ dx = Ω µ(x)|(-∆) s 2 u| q + λf (x) φ dx, ∀ φ ∈ X s , (4.1.9)
where X s is defined in (4.1.7).

Theorem 4.1.5. Assume that (B 1 ) holds. Then there exists λ * > 0 such that, for all 0 < λ ≤ λ * , (Q λ ) has a weak solution u ∈ W s,1 0 (Ω). Finally, for s ∈ (0, 1) and φ ∈ C ∞ 0 (R N ), following [START_REF] Ponce | Elliptic PDEs, measures and capacities. From the Poisson equations to nonlinear Thomas-Fermi problems[END_REF][START_REF] Shieh | On a new class of fractional partial differential equations[END_REF], we define the (distributional Riesz) fractional gradient of order s as the vector field ∇ s : R N → R given by

∇ s φ(x) := R N φ(x) -φ(y) |x -y| s x -y |x -y| dy |x -y| N , ∀ x ∈ R N .
(4.1.10)

Then we deal with the Dirichlet problem

       (-∆) s u = µ(x)|∇ s u| q + λf (x) , in Ω, u = 0 , in R N \ Ω. ( Q λ )
Remark 4.1.6. The notion of weak solution to ( Q λ ) has to be understood as in Definition 4.1.2.

Theorem 4.1.6. Assume that (B 1 ) holds. Then there exists λ * > 0 such that, for all 0 < λ ≤ λ * , ( Q λ ) has a weak solution u ∈ W s,1 0 (Ω).

We end this section describing the organization of the chapter. In Section 4.2, we introduce the suitable functional setting to deal with our problems and we also recall some known results that will be useful. In Section 4.3, which is independent of the rest of the chapter, we prove Calder ón-Zygmund type regularity results for the fractional Poisson equation ( 4

Functional setting and Useful tools

In this section we present the functional setting and some auxiliary results that will play an important role throughout the chapter. We begin recalling the definition of the fractional Sobolev space. Definition 4.2.1. Let Ω be an open set in R N and s ∈ (0, 1). For any p ∈ [1, ∞), the fractional Sobolev space W s,p (Ω) is defined as

W s,p (Ω) := u ∈ L p (Ω) : Ω×Ω |u(x) -u(y)| p |x -y| N +sp dxdy < ∞ .
It is a Banach space endowed with the usual norm Having at hand this definition we introduce the suitable space to deal with our problems.

Definition 4.2.2. Let Ω ⊂ R N be a bounded domain with boundary ∂Ω of class C 0,1 and s ∈ (0, 1). For any p ∈ [1, ∞). We define the space W s,p 0 (Ω) as

W s,p 0 (Ω) := u ∈ W s,p (R N ) : u = 0 in R N \ Ω .
It is a Banach space endowed with the norm

u W s,p 0 (Ω) := D Ω |u(x) -u(y)| p |x -y| N +sp dxdy 1/p , where D Ω := (R N × R N ) \ (CΩ × CΩ) = (Ω × R N ) ∪ (CΩ × Ω).
The space W s,p 0 (Ω) was first introduced in [START_REF] Servadei | Mountain pass solutions for non-local elliptic operators[END_REF] in the particular case p = 2. We refer to [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF] for more details on fractional Sobolev spaces. Nevertheless, due to their relevance in this work, we recall here some results involving fractional Sobolev spaces.

We shall make use of the following classical fractional Sobolev inequality. See [START_REF] Ponce | Elliptic PDEs, measures and capacities. From the Poisson equations to nonlinear Thomas-Fermi problems[END_REF]Proposition 15.5] for a beautiful proof. Theorem 4.2.1 (Sobolev inequality). For any s ∈ (0, 1), p ∈ [1, N s ) and u ∈ W s,p (R N ), it follows that

u L p * s (R N ) ≤ S N ,p R 2N |u(x) -u(y)| p |x -y| N +sp dxdy 1/p
, where S N ,p > 0 is a constant depending only on N and p.

Next, we present a fractional Hardy inequality and some of its consequences. These results will be crucial to show the optimality of the regularity assumptions of Theorem 4.1.1, namely to prove Theorem 4.1.3.

Theorem 4.2.2. [63, Theorem 1.1] Let N ≥ 2, 0 < s < 1 and p > 1. Then, for all u ∈ C ∞ 0 (R N ), it follows that R 2N |u(x) -u(y)| p |x -y| N +ps dxdy ≥ Λ N ,p,s R N |u(x)| p |x| ps dx, (4.2.1)
where

Λ N ,s,p := 2 1 0 σ ps-1 1 -σ N -ps p Φ N ,s,p (σ )dσ > 0, (4.2.2) and Φ N ,s,p (σ ) := |S N -2 | 1 -1 (1 -t 2 ) N -3 2 (1 -2σ t + σ 2 ) N +ps 2 dt. Proposition 4.2.3. Let Ω ⊂ R N , N ≥ 2,
Λ(Ω) := inf                    D Ω |φ(x) -φ(y)| p |x -y| N +ps dxdy Ω |φ(x)| p |x| ps dx : φ ∈ C ∞ 0 (Ω) \ {0}                    , it follows that Λ(Ω) = Λ N ,s,p where Λ N ,s,p > 0 is defined in (4.2.2).
2) The weight |x| -ps is optimal in the sense that, for all ε > 0, if follows that

inf                    D Ω |φ(x) -φ(y)| p |x -y| N +ps dxdy Ω |φ(x)| p |x| ps+ε dx : φ ∈ C ∞ 0 (Ω) \ {0}                    = 0.
Proof. Since the proof of 1) can be found in [3, Lemma 3.4], we just provide the proof of 2). Let ε > 0 be fixed but arbitrarily small. We assume by contradiction that there exists a smooth bounded 123 domain Ω ⊂ R N such that 0 ∈ Ω and

Λ ε (Ω) := inf                    D Ω |φ(x) -φ(y)| p |x -y| N +ps dxdy Ω |φ(x)| p |x| ps+ε dx : φ ∈ C ∞ 0 (Ω) \ {0}                    > 0. (4.2.3)
Let us then observe that for any B r (0) ⊂ Ω, it follows that

0 < Λ ε (Ω) ≤ Λ ε (B r (0)). ( 4
(B r (0)), 0 < Λ ε (Ω) ≤ Λ ε (B r (0)) ≤ D B r (0) |φ(x) -φ(y)| p |x -y| N +ps dxdy B r (0) |φ(x)| p |x| ps+ε dx ≤ r ε D B r (0) |φ(x) -φ(y)| p |x -y| N +ps dxdy B r (0) |φ(x)| p |x| ps dx .
Thus, by the definition of Λ(B r (0)) and 1), we deduce that 0

< Λ ε (Ω) r ε
≤ Λ B r (0) = Λ N ,s,p . Since (by assumption) Λ ε (Ω) > 0 and Λ N ,s,p is independent of Ω, letting r → 0, we obtain a contradiction and the result follows.

In order to prove some of the Calder ón-Zygmund type regularity results of Section 4.3, we will use the relation between the fractional Sobolev space W s,p (R N ) and the Bessel potential space defined below. Definition 4.2.3. Let s ∈ (0, 1). For any p ∈ [1, ∞), the Bessel potential space L s,p (R N ) is defined as

L s,p (R N ) := u ∈ L p (R N ) such that u = (I-d) -s 2 f with f ∈ L p (R N ) .
It is a Banach space endowed with the norm

|||u||| L s,p (R N ) := u L p (R N ) + f L p (R N ) .
Having in mind the fractional gradient of order s introduced in (4.1.10), let us point out that in [START_REF] Shieh | On a new class of fractional partial differential equations[END_REF]Theorem 1.7] it is proved that

L s,p (R N ) = u ∈ L p (R N ) such that |∇ s u| ∈ L p (R N ) , (4.2.6) with the equivalent norm u L s,p (R N ) := u L p (R N ) + ∇ s u L p (R N ) .
Notice also that in the case where s is an integer and 1 < p < ∞, by [START_REF] Adams | Sobolev spaces[END_REF]Theorem 7.63] we know that L s,p (R N ) = W s,p (R N ). Differently, in case s ∈ (0, 1), the two previous spaces does not coincide. However, we have the following result. Theorem 4.2.4. [9, Theorem 7.63] Assume that s ∈ (0, 1) and 1 < p < ∞. For all 0 < ε < s, it follows that

L s+ε,p (R N ) ⊂ W s,p (R N ) ⊂ L s-ε,p (R N ),
with continuous inclusions.

Finally, we recall a classical result of harmonic analysis that will be useful in Section 4.3.

Lemma 4.2.5. [109, Theorem I, Section 1.2, Chapter V] Let 0 < λ < N and 1 ≤ p < < ∞ be such

that 1 + 1 = 1 p + λ N . For g ∈ L p (R N ), we define J λ (g)(x) = R N g(y) |x -y| λ dy.
Then, it follows that: a) J λ is well defined in the sense that the integral converges absolutely for almost all

x ∈ R N . b) If p > 1, then J λ (g) L (R N ) ≤ c p,q g L p (R N ) . c) If p = 1, then {x ∈ R N |J λ (g)(x) > σ } ≤ A g L 1 (R N )
σ .

Regularity results for the fractional Poisson equation

The main goal of this section, which is independent of the rest of the work, is to prove sharp Calder ón-Zygmund type regularity results for the fractional Poisson equation

       (-∆) s v = h(x), in Ω, v = 0, in R N \ Ω, (4.3.1)
under the assumption

           Ω ⊂ R N , N ≥ 2, is a bounded domain with ∂Ω of class C 2 , s ∈ (1/2, 1), h ∈ L m (Ω) for some m ≥ 1. (4.3.2)
First of all, let us precise the notion of weak solution to (4.3.1).

Definition 4.3.1. We say that v is a weak solution to (4

.3.1) if v ∈ L 1 (Ω), v ≡ 0 in CΩ := R N \ Ω and Ω v(-∆) s φ dx = Ω h(x)φ dx , ∀ φ ∈ X s ,
where X s is defined in (4.1.7).

Under our assumption (4.3.2), the existence and uniqueness of solutions to (4.3.1) is a particular case of [START_REF] Chen | Semilinear fractional elliptic equations involving measures[END_REF]Proposition 2.4] (see also [START_REF] Leonori | Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations[END_REF]Section 4]). Having this in mind, we prove several regularity results for (4.3.1). Our first main result reads as follows: Proposition 4.3.1. Assume (4.3.2) and let v be the unique weak solution to (4.3.1) and t ∈ (0, 1):

1) If m = 1, then v ∈ W t,p 0 (Ω) for all 1 ≤ p < N N -(2s-t) and there exists C 1 = C 1 (s, t, p, Ω) > 0 such that v W t,p 0 (Ω) ≤ v W t,p (R N ) ≤ C 1 h L 1 (Ω) . 2) If 1 < m < N 2s , then v ∈ W t,p 0 (Ω) for all 1 ≤ p ≤ mN N -m(2s-t) and there exists C 1 = C 1 (m, s, t, p, Ω) > 0 such that v W t,p 0 (Ω) ≤ v W t,p (R N ) ≤ C 1 h L m (Ω) . 3) If N 2s ≤ m < N 2s-1 , then v ∈ W t,p 0 (Ω) for all 1 ≤ p < mN t(N -m(2s-1)
) and there exists

C 1 = C 1 (m, s, t, p, Ω) > 0 such that v W s,p 0 (Ω) ≤ v W t,p (R N ) ≤ C 1 h L m (Ω) . 4) If m ≥ N 2s-1 , then v ∈ W t,p 0 (Ω) for all 1 ≤ p < ∞. Remark 4.3.1.
a) The previous results are sharp in the sense that, if "we take t = s = 1", we recover the classical sharp regularity results for the local case and those cannot be improved. See for instance [START_REF] Ponce | Elliptic PDEs, measures and capacities. From the Poisson equations to nonlinear Thomas-Fermi problems[END_REF]Chapter 5]. b) In the particular case of the fractional Laplacian of order s ∈ (1/2, 1) and for h ∈ L 1 (Ω), we improve the regularity results of [START_REF] Abdellaoui | On the fractional p-Laplacian equations with weight and general datum[END_REF][START_REF] Kuusi | Nonlocal equations with measure data[END_REF][START_REF] Leonori | Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations[END_REF]. Note however that in the three quoted papers the authors deal with more general operators and cover the full range s ∈ (0, 1). Furthermore, in [START_REF] Kuusi | Nonlocal equations with measure data[END_REF] the authors also deal with measures as data. c) Since s ∈ (1/2, 1), observe that t < 2s for all t ∈ (0, 1).

As we believe it has its own interest, let us highlight a particular case of the previous result.

1) If m = 1, then v ∈ W s,p 0 (Ω) for all 1 ≤ p < N N -s and there exists C 1 = C 1 (s, p, Ω) > 0 such that v W s,p 0 (Ω) ≤ v W s,p (R N ) ≤ C 1 h L 1 (Ω) . 2) If 1 < m < N 2s , then v ∈ W s,p 0 (Ω) for all 1 ≤ p ≤ mN N -ms and there exists C 1 = C 1 (m, s, p, Ω) > 0 such that v W s,p 0 (Ω) ≤ v W s,p (R N ) ≤ C 1 h L m (Ω) . 3) If N 2s ≤ m < N 2s-1 , then v ∈ W s,p 0 (Ω) for all 1 ≤ p < mN s(N -m(2s-1)
) and there exists

C 1 = C 1 (m, s, p, Ω) > 0 such that v W s,p 0 (Ω) ≤ v W s,p (R N ) ≤ C 1 h L m (Ω) . 4) If m ≥ N 2s-1 , then v ∈ W s,p 0 (Ω) for all 1 ≤ p < ∞.
In the following two results we complete the information obtained in Proposition 4.3.1 when h ∈ L m (Ω) for some m > N /2s. Proposition 4.3.3. Assume (4.3.2) and let v be the unique weak solution to (4.3.1) and t ∈ (0, s): Due to the nonlocality of the fractional Laplacian, several notions of regularity can be studied. The following results, which generalize the fractional regularity proved in [84, Theorem 24] with a different approach, can be seen as the counterpart of Proposition 4.3.1 to deal with (Q λ ) and ( Q λ ). Proposition 4.3.5. Assume (4.3.2) and let v be the unique weak solution to (4.3.1) and t ∈ (0, s]:

1) If N 2s ≤ m < N 2s-t then v ∈ W t,p 0 (Ω) for all 1 ≤ p < mN N -m(2s-t) and there exists C 2 = C 2 (m, s, t, p, Ω) > 0 such that v W t,p 0 (Ω) ≤ v W t,p (R N ) ≤ C 2 h L m (Ω) . 2) If m ≥ N 2s-t then v ∈ W t,p 0 (Ω) for all 1 ≤ p < ∞ and there exists C 2 = C 2 (m, s, t, p, Ω) > 0 such that v W t,p 0 (Ω) ≤ v W t,p (R N ) ≤ C 2 h L m (Ω
1) If m = 1, then (-∆) t 2 v ∈ L p (Ω) for all 1 ≤ p < N N -(2s-t) and there exists C 3 = C 3 (s, t, p, Ω) > 0 such that (-∆) t 2 v L p (Ω) ≤ C 3 h L 1 (Ω) . 2) If 1 < m < N 2s-t , then (-∆) t 2 v ∈ L p (Ω) for all 1 ≤ p ≤ mN N -m(2s-t) and there exists C 3 = C 3 (s, t, m, p, Ω) > 0 such that (-∆) t 2 v L p (Ω) ≤ C 3 h L m (Ω) . 3) If m ≥ N 2s-t then (-∆) t 2 v ∈ L p (Ω) for all 1 ≤ p < ∞ and there exists C 3 = C 3 (s, t, m, p, Ω) > 0 such that (-∆) t 2 v L p (Ω) ≤ C 3 h L m (Ω) .
∇ s v L p (Ω) ≤ C 4 h L 1 (Ω) .
2) If 1 < m < N s , then |∇ s v| ∈ L p (Ω) for all 1 ≤ p ≤ mN N -ms and there exists Theorems 13,[START_REF] Arcoya | Regularizing effect of the interplay between coefficients in some elliptic equations[END_REF][START_REF]Regularizing effect of L q interplay between coefficients in some elliptic equations[END_REF][START_REF] Boccardo | Existence de solutions faibles pour des équations elliptiques quasilinéaires à croissance quadratique[END_REF][START_REF]Quelques propriétés des opérateurs elliptiques quasi linéaires[END_REF] Let Ω ⊂ R N , N ≥ 2, be a bounded domain with boundary ∂Ω of class C 0,1 , let s ∈ (0, 1) and assume that h ∈ L m (Ω) for some m ≥ 1. Then problem (4.3.1) has an unique weak solution. Moreover:

C 4 = C 4 (s, m, p, Ω) > 0 such that ∇ s v L p (Ω) ≤ C 4 h L m (Ω) . 3) If m ≥ N s then |∇ s v| ∈ L p (Ω)
1) If m = 1, then v ∈ L p (Ω) for all 1 ≤ p < N N -2s and there exists C 5 = C 5 (s, p, Ω) > 0 such that v L p (Ω) ≤ C 5 h L 1 (Ω) . 2) If 1 < m < N 2s , then v ∈ L p (Ω) for all 1 ≤ p ≤ mN N -2ms and there exists C 5 = C 5 (s, m, p, Ω) > 0 such that v L p (Ω) ≤ C 5 h L m (Ω) . 3) If m ≥ N 2s , then v ∈ L p (Ω) for all 1 ≤ p < ∞ and there exists C 5 = C 5 (s, m, p, Ω) > 0 such that v L p (Ω) ≤ C 5 h L m (Ω) .
Considering stronger assumptions, the first author and I. Peral proved in [START_REF]Towards a deterministic KPZ equation with fractional diffusion: the stationary problem[END_REF] that the unique weak solution to (4.3.1) belongs to a suitable local Sobolev space. More precisely, under the assumption (4.3.2) the authors obtained the following result.

v W 1,p (R N ) ≤ C 6 ∇v L p (Ω) ≤ C 6 h L 1 (Ω) . 2) If 1 < m < N 2s-1 , then v ∈ W 1,p (R N ) for all 1 ≤ p ≤ mN N -m(2s-1) and there exists C 6 = C 6 (m, s, p, Ω) > 0 such that v W 1,p (R N ) ≤ C 6 ∇v L p (Ω) ≤ C 6 h L m (Ω) . 3) If m ≥ N 2s-1 , then v ∈ W 1,p (R N ) for all 1 ≤ p < ∞ and there exists C 6 = C 6 (m, s, p, Ω) > 0 such that v W 1,p (R N ) ≤ C 6 ∇v L p (Ω) ≤ C 6 h L m (Ω) .
As last ingredient to prove our regularity results we need an interpolation result that we borrow from [START_REF] Brezis | Gagliardo-Nirenberg inequalities and non-inequalities: The full story[END_REF]. Let us introduce the real numbers 0 ≤ s 1 ≤ η ≤ s 2 ≤ 1 and 1 ≤ p 1 , p 2 , p ≤ ∞ and assume that they satisfy the relations Having at hand all the needed ingredients, we prove our first regularity result.

η = θs 1 + (1 -θ)s 2 and 1 p = θ p 1 + 1 -θ p 2 with 0 < θ < 1. ( 4
w W η,p (R N ) ≤ C w θ W s 1 ,p 1 (R N ) w 1-θ W s 2 ,p 2 (R N ) , ∀ w ∈ W s 1 ,p 1 (R N ) ∩ W s 2 ,p 2 (R N ) .
Proof of Proposition 4.3.1. 1) Let v be the unique weak solution to (4.3.1). On the one hand, by Lemma 4.3.7, 1), we know that

v L p 1 (R N ) ≤ C 5 h L 1 (Ω) , ∀ 1 ≤ p 1 < N N -2s . (4.3.5)
On the other hand, by Lemma 4.3.8, 1), we know that

v W 1,p 2 (R N ) ≤ C 6 h L 1 (Ω) , ∀ 1 ≤ p 2 < N N -(2s -1) . (4.3.6)
Also, by Lemma 4.3.9 applied with η = t, s 1 = 0 and s 2 = 1, we have that

v W t,p (R N ) ≤ C v 1-t L p 1 (R N ) v t W 1,p 2 (R N ) . ( 4.3.7)
The result follows from (4.3.5)-(4.3.7) using that

1 ≥ 1 p = 1 -t p 1 + t p 2 > (1 -t)(N -2s) + t(N -(2s -1)) N = N -(2s -t) N .
2) Let v be the unique weak solution to (4.3.1). By Lemma 4.3.7, 2), we know that

v L p 1 (R N ) ≤ C 5 h L m (Ω) , ∀ 1 ≤ p 1 ≤ mN N -2ms . ( 4
v W 1,p 2 (R N ) ≤ C 6 h L m (Ω) , ∀ 1 ≤ p 2 ≤ mN N -m(2s -1) . (4.3.9)
Finally, by Lemma 4.3.9 applied with η = t, s 1 = 0 and s 2 = 1, we know that (4.3.7) holds. The result follows from (4.3.7), (4.3.8) and (4.3.9) using that

1 ≥ 1 p = 1 -t p 1 + t p 2 ≥ (1 -t)(N -2ms) + t(N -m(2s -1)) mN = N -m(2s -t) mN .
3) Let v be the unique weak solution to (4.3.1). By Lemma 4.3.7, 3), we know that

v L p 1 (R N ) ≤ C 5 h L m (Ω) , ∀ 1 ≤ p 1 < ∞. ( 4
1) If m = 1, then v ∈ L s,p (R N ) for all 1 ≤ p < N N -s and there exists C 7 = C 7 (s, p, Ω) > 0 such that v L s,p (R N ) ≤ C 7 h L 1 (Ω) . 2) If 1 < m < N s , then v ∈ L s,p (R N ) for all 1 ≤ p ≤ mN N -ms and there exists C 7 = C 7 (m, s, p, Ω) > 0 such that v L s,p (R N ) ≤ C 7 h L m (Ω) .
3) If m ≥ N s , then v ∈ L s,p (R N ) for all 1 ≤ p < ∞ and there exists

C 7 = C 7 (m, s, p, Ω) > 0 such that v L s,p (R N ) ≤ C 7 h L m (Ω) .
Proof. Assume (4.3.2) and let v be the unique weak solution to (4.3.1). Taking into account (4.2.6), we have just to show the regularity of |∇ s v| where ∇ s is defined in (4.1.10). By a density argument and [95, Lemma 15.9] we have that

|∇ s v(x)| ≤ 1 N -(1 -s) R N |∇v(y)| |x -y| N -(1-s) dy a.e. in R N . (4.3.12)
Let us then split into three cases:

1) m = 1.

By Lemma 4.3.8, 1), we get v ∈ W 1,q (R N ) for all 1 ≤ q < N N -(2s-1) . Thus, by Lemma 4.2.5, we conclude that |∇ s v(x)| ∈ L p (R N ) for all 1 ≤ p < N N -s .

2) 1 < m < N s . The result follows arguing on the same way, using now (4.3.12), Lemma 4.3.8, 2) and Lemma 4.2.5.

3) m ≥ N s . In this case, since m ≥ N s and Ω is a bounded domain then f ∈ L m(Ω) for all m < N s . In particular, using the second point it follows that v ∈ L s, p(R N ) for all 1 ≤ p ≤ mN N -s m . Letting m ↑ N s , we reach that p ↑ ∞. Hence v ∈ L s,p (R N ) for all 1 ≤ p < ∞. Now, using the above regularity result in the Bessel potential space, we obtain the following lemma.

Lemma 4.3.12. Assume (4.3.2) and let v be the unique weak solution to (4.3.1):

1) If N 2s ≤ m < N s then v ∈ W s ,p
0 (Ω) for all 0 < s < s and all 1 ≤ p ≤ mN N -ms . Moreover, there exists

C 8 = C 8 (m, s , p, Ω) > 0 such that v W s ,p 0 (Ω) ≤ v W s ,p (R N ) ≤ C 8 h L m (Ω) .
2) If m ≥ N s , then v ∈ W s ,p 0 (Ω) for all 0 < s < s and all 1 ≤ p < ∞. Moreover, there exists

C 8 = C 8 (m, s , p, Ω) > 0 such that v W s ,p 0 (Ω) ≤ v W s ,p (R N ) ≤ C 8 h L m (Ω) .
Proof. First observe that without loss of generality we can assume that p > 1. For p = 1 the result follows from Proposition 4.3.1 and the continuous embedding W s,p (R N ) ⊂ W s ,p (R N ). We consider then two cases:

) N 2s ≤ m < N 2s-t .
Let v be the unique weak solution to (4.3.1). On the one hand, by Lemma 4.3.12, 1), we have that v ∈ W s ,p 1 (R N ) for all 0 < s < s and all 1 ≤ p 1 ≤ mN N -ms and that there exists

C 8 > 0 such that v W s ,p 1 0 (Ω) ≤ v W s ,p 1 (R N ) ≤ C 8 h L m (Ω) . (4.3.13)
On the other hand, by Lemma 4.3.10, we have that v ∈ W η,q 1 (R N ) for all 0 < η < s < s and all

1 ≤ q 1 ≤ N p 1 N -p 1 (s -η) ≤ mN N -m(s+s -η) . Moreover, there exists C > 0 such that v W η,q 1 (R N ) ≤ C v W s ,p 1 (R N ) ≤ C C 8 h L m (Ω) . (4.3.14)
We fix then p ∈ [1, mN N -m(2s-t) ) and observe that we can find η ∈ (t, s ) such that

1 ≤ p ≤ mN N -m(s + s -η) .
The result follows from (4.3.14) using the continuous embedding W η,p (R N ) ⊂ W t,p (R N ).

2) m ≥ N 2s-t . The result follows arguing as in the proof of Proposition 4.3.11, 3) using Proposition 4.3.3, 1).

Proof of Proposition 4.3.4. On the one hand, by Lemma 4.3.12, 1), we have that

v W s ,p 1 0 (Ω) ≤ v W s ,p 1 (R N ) ≤ C 8 h L m (Ω) , ∀ 0 < s < s, ∀ 1 ≤ p 1 ≤ mN N -ms . (4.3.15)
On the other hand, by Lemma 4.3.8, 2), it follows that

v W 1,p 2 (R N ) ≤ C 6 h L m (Ω) , ∀ 1 ≤ p 2 ≤ mN N -m(2s -1) . (4.3.16)
Also, by Lemma 4.3.9 applied with η = t, s 1 = s and s 2 = 1, we know that

v W t,p (R N ) ≤ C v 1-t 1-s W s ,p 1 (R N ) v t-s 1-s W 1,p 2 (R N ) , (4.3.17)
with 1 p = 1 1 -s 1 -t p 1 + t -s p 2 .
We fix then an arbitrary 1 ≤ p < mN N -m(2s-t) and observe that we can choose s < s such that p = p. Hence, the result follows from (4.3.15)-(4.3.17). Then, observe that, for all x ∈ Ω,

|(-∆) t 2 v(x)| ≤ R N |v(x) -v(y)| |x -y| N +t dy ≤ S 1 |v(x) -v(y)| |x -y| N +t dy + S 2 |v(x) -v(y)| |x -y| N +t dy + B 1 (x) |v(x) -v(y)| |x -y| N +t dy =: I 1 (x) + I 2 (x) + I 3 (x). (4.3.18)
Now, let us estimate each one of the three terms. First observe that

I 1 (x) = S 1 |v(x)| |x -y| N +t dy ≤ S 1 |v(x)| dist(y, Ω) N +t dy = |v(x)| S 1 dy dist(y, Ω) N +t dy = c 1 (N , t, Ω)|v(x)|, ∀ x ∈ Ω. (4.3.19)
Next, using that Ω is a bounded domain and the triangular inequality, we deduce that

I 2 (x) ≤ S 2 |v(x) -v(y)|dy ≤ c 2 (Ω)|v(x)| + v L 1 (Ω) , ∀ x ∈ Ω. (4.3.20)
Finally, following the arguments of [55, Proposition 2.2], we deduce that

I 3 (x) = B 1 (0) |v(x) -v(x + z)| |z| 1 |z| N +t-1 dz = B 1 (0) 1 0 |∇v(x + τz)| |z| N +t-1 dτdz ≤ 1 0 R N |∇v(w)| |w -x| N +t-1 τ t-1 dwdτ = R N |∇v(w)| |w -x| N +t-1 dw 1 0 τ t-1 dτ = 1 t R N |∇v(w)| |w -x| N +t-1 dw, ∀ x ∈ Ω. ( 4
|(-∆) t 2 v(x)| ≤ c(s, t, Ω) |v(x)| + R N |∇v(w)| |w -x| N +t-1 dw + v L 1 (Ω) , ∀ x ∈ Ω, (4.3.22)
and so, exploiting again the fact that Ω is a bounded domain and using the Hölder and triangular inequalities, for all 1 ≤ p < ∞, we obtain that (-∆)

t 2 v L p (Ω) ≤ c 2 (s, t, Ω)       v L p (Ω) + R N |∇v(w)| |w -x| N +t-1 dw L p (R N )       . (4.3.23)
Now, let us split the proof into three parts:

1) m = 1.

By Lemma 4.3.8, 1), we know that v ∈ W 1,σ 0 (Ω) for all 1 ≤ σ < N N -(2s-1) and there exists

C 6 = C 6 (s, σ , Ω) > 0 such that ∇v L σ (R N ) ≤ C 6 h L 1 (Ω) , ∀ 1 ≤ σ < N N -(2s -1)
.

Thus, applying Lemma 4.2.5, we deduce that

R N |∇v(w)| |w -x| N +t-1 dw L (R N ) ≤ C C 6 h L 1 (Ω) , ∀ 1 ≤ < N N -(2s -t) . ( 4

.3.24)

Also, by Lemma 4.3.7, 1), we know that

v L γ (Ω) ≤ C 5 h L 1 (Ω) , ∀ 1 ≤ γ < N N -2s . ( 4
= C 6 (m, s, σ , Ω) > 0 such that ∇v L σ (R N ) ≤ C 6 h L m (Ω) , ∀ 1 ≤ σ ≤ mN N -m(2s -1)
.

Hence, by Lemma 4.2.5, we deduce that

R N |∇v(w)| |w -x| N +t-1 dw L (R N ) ≤ C C 6 h L m (Ω) , ∀ 1 ≤ ≤ mN N -m(2s -t) . ( 4
v L γ (Ω) ≤ C 5 h L m (Ω) , ∀ 1 ≤ γ < mN N -2ms , if 1 ≤ m < N 2s (4.3.27) and v L γ (Ω) ≤ C 5 h L m (Ω) , ∀ 1 ≤ γ < ∞, if N 2s ≤ m < N 2s -1 . ( 4
∇ s u(x) = 1 N -(1 -s) R N ∇u(y)
|x -y| N +s-1) dy. The result then follows arguing as in the proof of Proposition 4.3.5.

Convergence and compactness

We end this section presenting a result of convergence and one of compactness for the fractional Poisson equation (4.3.1). They will be used in the proofs of Theorems 4.1.1, 4.1.4, 4.1.5 and 4.1.6. Proposition 4.3.13. Let Ω ⊂ R N , N ≥ 2, be a bounded domain with ∂Ω of class C 2 , let s ∈ (1/2, 1), let {h n } ⊂ L 1 (Ω) be a sequence such that h n → h in L 1 (Ω) and let v n be the unique weak solution to

       (-∆) s v n = h n (x),
in Ω,

v n = 0, in R N \ Ω,
for all n ∈ N, and v be the unique weak solution to Hence, we conclude that g has exactly one root t * ∈ (0, ∞). Let us begin recalling that, under the assumption (A 1 ), f ∈ L m (Ω) for some m > N 2s . Hence, since we are working in a bounded domain, without loss of generality, we can assume that m ∈ ( N 2s , N 2s-1 ). Moreover, observe that, for λf ≡ 0, u ≡ 0 is a solution to (P λ ) and, for µ ≡ 0, (P λ ) reduces to (4.3.1). Hence, we may assume that µ ∞ 0 and f L m (Ω) 0.

       (-∆) s v = h(x), in Ω, v = 0, in R N \ Ω. Then v n → v in W
Next, we fix some notation that we use throughout this subsection. First, we fix r = r(m, s) > 0 such that 1 < 2m < r < mN s(Nm(2s -1))

, and ε = ε(r, m, s) > 0 such that 1 < r < mN (s + ε)(Nm(2s -1)) < mN s(Nm(2s -1))

, s + ε < 1, and sε > 1 2 .

Also, we introduce and fix the constants Taking into account the above inequality, if we show that I 1 is bounded and I 2 goes to zero, we deduce that D 2 s (ϕ n ) -D 2 s (ϕ) L 1 (Ω) → 0. Claim 1: I 1 is bounded. (4.4.12)

Directly observe that

Since ϕ n , ϕ ∈ E for all n ∈ N, by (4.4.7), we have that

D 2 s (ϕ n ) L 1 (Ω) + D 2 s (ϕ) L 1 (Ω) ≤ 2c C 2 9 l < ∞,
and so, that I 1 is bounded.

Claim 2: I 2 goes to zero.

Let θ ∈ (0, 1) be small enough to ensure that 2-θ 1-θ < r. By Hölder inequality, it follows that where C, C and C are positive constants independent of n. Thus, we conclude that (4.4.14) is indeed bounded.

D 2 s (ϕ n -ϕ) L 1 (Ω) = Ω R N
From Claim 1 and 2 we deduce that D 2 s (ϕ n ) -D 2 s (ϕ) L 1 (Ω) → 0. This implies that g n → g in L 1 (Ω), as desired, and the result follows. Lemma 4.4.6. Assume that (A 1 ) holds. Then T is compact.

Proof. Let {ϕ n } ⊂ E be a bounded sequence in W s,1 0 (Ω) and define u n = T (ϕ n ) for all n ∈ N. We have to show that u n → u in W s,1 0 (Ω) for some u ∈ W s,1 0 (Ω). Since {ϕ n } ⊂ E for all n ∈ N, arguing as in Lemma 4.4.3, we deduce that {D 2 s (ϕ n )} is a bounded sequence in L 1 (Ω). Hence, if we define g n (x) := D 2 s (ϕ n ) + λf (x), ∀ n ∈ N, we have that {g n } is a bounded sequence in L 1 (Ω). The result then follows from Proposition 4.3.14.

Proof. First observe that, with the above notation, we have that Thus, as by Proposition 4.3.1 we also know that u ∈ W s,1 0 (Ω), we conclude that u ∈ E 1 and so, that T 1 (E 1 ) ⊂ E 1 .

Lemma 4.4.10. Assume (A 1 ). Then T 1 is continuous.

Proof. Let {ϕ n } ⊂ E be a sequence such that ϕ n → ϕ in W s,1 0 (Ω) and define u n = T 1 (ϕ n ), for all n ∈ N, and u = T 1 (ϕ). Arguing as in the proof of Lemma 4.4.5, we just have to prove that Then, arguing exactly as in Lemma 4.4.5 and using that ϕ n ∞ ≤ C (independent of n) we deduce that I 1 → 0. On the other hand, we know that D 2 s (ϕ) L m (Ω) < ∞. Hence, since ϕ n → ϕ in L α (Ω) for all 1 ≤ α < ∞, we also obtain that I 2 → 0. We then conclude that (4.4.22) holds, as desired.

Proof of Theorem 4. 1.4. Observe that the compactness of T 1 follows arguing exactly as in Lemma 4.4.6. Hence, since E 1 is a closed convex set of W s,1 0 (Ω) and, by Lemmas 4.4.9, 4.4.8 and 4.4.10 we know that T 1 is well defined, continuous and satisfies T 1 (E 1 ) ⊂ E 1 , we can apply the Schauder fixed point Theorem to obtain u ∈ E 1 such that T 1 (u) = u. Thus, we conclude that ( P λ ) has a weak solution for all 0 < λ ≤ λ * . Finally, since u ∈ W s,1 0 (Ω) ∩ W s,r 0 (Ω) for some 1 < 2 < r, by Lemma 4.3.9 we deduce that u ∈ W s,2 0 (Ω). Moreover, since r > N /s, by [55, Theorem 8.2], we know that every ϕ ∈ E 1 belongs to C 0,α (Ω) for some α > 0. for some ε ∈ (0, 1) to be chosen later and observe that, since Ω is bounded, f ∈ L m (Ω).

We assume by contradiction that, for all ε > 0, there exists λ ε > 0 such that (P λ ) has a solution u ∈ W s,2 0 (Ω). Arguing as in the proof of Theorem 4.1.2, we conclude that, for all φ ∈ C ∞ 0 (Ω) \ {0}, Thus, we deduce that This section is devoted to the proofs of Theorems 4.1.5 and 4.1.6. First, having at hand Proposition 4.3.5, we prove Theorem 4.1.5 using again a fixed point argument. The proof is similar to the ones performed in Section 4.4. Hence, we skip some details.

0 < µ 1 λ ε inf                  D Ω |φ(x) -φ(y)| 2 |x -y| N +2s dydx Ω φ 2 (x) |x| N -ε m dx : φ ∈ C ∞ 0 (Ω) \ {0}                  . ( 4
Since Ω is bounded, without loss of generality, we assume that 1 ≤ m < N s . Also, if λf ≡ 0, it follows that u ≡ 0 is a solution to (Q λ ) and, if µ ≡ 0, (Q λ ) reduces to (4.3.1). Hence, we may also assume that µ ∞ 0 and f L m (Ω) 0.

Next, we fix some notation that will be used throughout the section. First, we fix r = r(m, s, q) > 0 such that 1 < qm < r < mN Nms , C 3 the constant given by Proposition 4.3.5 with p = r and

λ * = q -1 q f L m (Ω)       1 qC q 3 |Ω| r-qm r µ ∞       1 q-1 .
Then, by the definition of λ * and Lemma 4. With the above constants fixed, we introduce

E 2 := v ∈ W s,1 0 (Ω) : (-∆) s 2 v L r (Ω) ≤ l 1 q ,
and observe that E 2 is a closed convex set of W s,1 0 (Ω). Then, we define T 2 : E 2 → W s,1 0 (Ω) by T 2 (ϕ) = u with u the unique weak solution to

       (-∆) s u = µ(x)|(-∆) s 2 ϕ| q + λf (x) ,
in Ω,

u = 0 , in R N \ Ω, (4.6.2)
and observe that (Q λ ) is equivalent to the fixed point problem u = T 2 (u). Hence, we shall show that T 2 has a fixed point. Proof. Let {ϕ n } ⊂ E 2 be a sequence such that ϕ n → ϕ in W s,1 0 (Ω) and define u n = T 2 (ϕ n ), for all n ∈ N, and u = T 2 (ϕ). We shall show that u n → u in W s,1 0 (Ω). Observe that

w n = u n -u satisfies        (-∆) s w n = µ(x) |(-∆) s 2 ϕ n | q -|(-∆) s 2 ϕ| q ,
in Ω, Since ϕ n → ϕ in W s,1 0 (Ω), it follows that ϕ nϕ → 0 almost everywhere in Ω. Furthermore, observe that, for all measurable subset ω ⊂ Ω, we have that

w n = 0 , in R N \ Ω.
ω |(-∆) s 2 (ϕ n -ϕ)| q dx ≤ 2 l |ω| r-q q .
Hence, by Vitali's convergence Theorem, (4.6.6) holds and the result follows.

Proof of Theorem 4.1.5. Since E 2 is a closed convex set of W s,1 0 (Ω) and, by Lemmas 4.6.1 and 4.6.2, we know that T 2 is continuous, compact and satisfies T 2 (E 2 ) ⊂ E 2 , we can apply the Schauder fixed point Theorem to obtain u ∈ E 2 such that T 2 (u) = u. Thus, we conclude that (Q λ ) has a weak solution for all 0 < λ ≤ λ * .

Proof of Theorem 4.1.6. Having at hand Corollary 4.3.6, the result follows arguing as in Theorem 4.1.5. can be tackled within the transformed variational framework set up in Chapter 1 and by using topological-variational methods in the presence of invariant sets.

The first preliminary step we plan to do is the study of the radial case (already interesting in itself). More precisely, assuming that c, µ and h are radial functions, we will address the problem where B R (0) := {x ∈ R N : |x| < R}. Our aim is to obtain a more precise information on the structure of the set of solutions to (P R λ ) for the non-coercive case λc 0. In particular, we wish to characterize branches of radial solutions (depending on λ) and to study whether bifurcation of nonradial solutions may occur along the branch. In such way, we expect to have a bigger intuition with regard to the general problem (P λ ) which should allow us to address the desired question of higher multiplicity.

Another challenging question is the existence of possible limiting and concentration behaviour of solutions to (P λ ) as λ tends to +∞. We expect that local extrema of the data functions c, µ should play a significant role with regard to the determination of possible concentration points. This expectation is based on the fact that a rescaling transformation of the form w(x) = u(x 0 + x √ λ

) leads to the equation

           -∆w = c(x 0 + x √ λ )w + µ(x 0 + x √ λ )|∇w| 2 + h(x 0 + x √ λ ) λ ,
in Ω λ , w = 0 , on ∂Ω λ , on a rescaled domain Ω λ . In the limit λ → +∞, an equation with constant coefficients c(x 0 ) and µ(x 0 ) is obtained. This equation then again fits in the variational framework set up in Chapter 1.

Using solutions to the limiting equation as building blocks, we wish to set of a Lyapunov-Schmidt type reduction framework to derive the existence of solutions to (P λ ) for large λ with one or more concentration points. In various cases, we expect new results on the multiplicity of solutions to (P λ ) for large λ. In the very recent preprint [START_REF]Global integrability and boundary estimates for uniformly elliptic PDE in divergence form[END_REF], B. Sirakov found the optimal ε for uniformly elliptic PDE in divergence form. More precisely, he deals with equations of the form div(A(x)∇u) + b(x)|∇u| + f (x) = 0, u ∈ H 1 0 (Ω), (P BH-2 )

where A is a symmetric matrix, b and f are non-negative functions and, for some λ > 0 and p > N , A ≥ λI, A ∈ W 1,p (Ω), b, f ∈ L p (Ω).

He proves the following result. Having in mind both results, we would like to obtain the corresponding optimal ε for the nonuniformly elliptic case of the p-Laplacian, namely problem (P BHP ). The proof of B. Sirakov is based on a not at all trivial Moser-type iteration. We hope to be able to perform this kind of iteration in the more general case of the p-Laplacian. Nevertheless, it is worth to point out that, since the p-Laplacian is a non-linear and non-uniformly elliptic operator, several difficulties are expected to appear and some new ideas will be needed. Even more generally, we would like to deal with a boundary value problem of the form div(A A A(x, u, ∇u)) + B B B(x, u, ∇u) = 0, u ∈ W

Nonlocal elliptic PDEs

Let Ω ⊂ R N , N ≥ 2, be a smooth bounded domain and s ∈ (1/2, 1). In Section 4.3, Chapter 4, we obtain sharp Calder ón-Zygmund type regularity results for the fractional Poisson equation

       (-∆) s u = f (x) ,
in Ω,

u = 0 , in R N \ Ω, (P f )
with f ∈ L m (Ω) for some m ≥ 1. Nevertheless, we shall observe that our results do not cover the full range s ∈ (0, 1). It will be our aim here to extend this results to the full range s ∈ (0, 1) and more general operators. We hope here to perform a purely non-local proof and cover the full range s ∈ (0, 1).

In Chapter 4, these regularity results are then used to obtain existence and non-existence results to a problem of the form

       (-∆) s u = µ(x) D 2 s (u) + λf (x) , in Ω, u = 0 , in R N \ Ω, (P N λ )
where λ > 0 is a real parameter, f belongs to a suitable Lebesgue space, µ ∈ L ∞ (Ω) and D 2 s is a nonlocal "gradient square" term given by This problem can be seen as the nonlocal counterpart of (P λ ) in the case where c ≡ 0. In comparison with the local problem (P λ ), very little is known about the nonlocal case. To be more precise, let us introduce the Dirichlet problem

       (-∆) s u = c(x)u + µ(x) D 2 s (u) + λf (x) , in Ω, u = 0 , in R N \ Ω,
where λ > 0 is a real parameter, c and f belongs to a suitable Lebesgue space and µ ∈ L ∞ (Ω). Having in mind the known results for the local case (P λ ), it seems interesting to address the following questions:

1) Does the uniqueness of (smooth) solutions hold for c(x) ≤ 0 ?

2) Under the assumption c(x) ≤ α 0 < 0 a.e. in Ω. Is it possible to remove the smallness condition on λ imposed on λ?

3) Is it possible to prove the existence of more than one solution for c(x) 0, µ(x) ≥ µ 1 > 0 and λ > 0 small enough ?

5 )

 5 For p ∈ [1, +∞[, the norm ( Ω |u| p dx) 1/p in L p (Ω) is denoted by • p . We denote by p the conjugate exponent of p, namely p = p/(p -1) and by p * the Sobolev critical exponent i.e. p * = N p N -p if p < N and p * = +∞ in case p ≥ N . The norm in L ∞ (Ω) is u ∞ = esssup x∈Ω |u(x)|. 6) The space W 1,p 0 (Ω) is equipped with the norm u W 1,p 0 (Ω) := Ω |∇u| p dx 1/p .

( 10 )

 10 Remark 0.1.2. a) We refer to [49, Definition I-3.1] for the definition of L p -Carathéodory we use here.

Figure 1 :

 1 Figure 1: Illustration of Theorem 0.1.7

Figure 2 :

 2 Figure 2: Existence regions of Theorem 0.1.8

  2.1. a) For a definition of Supp(f ) with f ∈ L p (Ω), for some p ≥ 1, we refer to [26, Proposition 4.17]. b) The condition c -≡ 0 in {x ∈ Ω : d(x, Ω + ) < ε} for some ε > 0 is reminiscent of the so-called "thick zero set" condition first introduced in [10]. c) Under the regularity assumptions of (22), any solution to (21) belongs to C 0,τ (Ω) for some τ > 0. This can be deduced from [82, Theorem IX-2.2] (see also [17, Proposition 2.1]).

Figure 4 :

 4 Figure 4: Illustration of Theorems 0.2.1 and 0.2.2

Figure 5 :

 5 Figure 5: Localization process

Figure 6 :

 6 Figure 6: Zoom around x

Theorem 0. 3 . 1 .

 31 [Theorem 3.1.1, Chapter 3] Assume that (32) holds. Then (31) has a solution if, and only if, m d > 0. Remark 0.3.2. a) This theorem generalizes [18, Proposition 3.1 and Remark 3.2] and, for p = 2, Theorem 0.1.4. b) By [17, Theorem 1.1] we know that the solution obtained is unique.

Figure 7 :

 7 Figure 7: Illustration of Theorems 0.3.4 and 0.3.5

  Remark 0.3.6. a) The case where B ∈ (L ∞ (Ω)) N and c ∈ L ∞ (Ω) is nowadays classical and can be founded for instance in [112, Theorem 3.27]. b) Theorem 0.3.7 can be obtained as a corollary from [100, Theorem 4.1]. Nevertheless, for the benefit of the reader, we provide a self-contained simplified proof in Chapter 3, Appendix 3.7.

( 47 )

 47 Theorem 0.4.5. [Theorem 4.1.5, Chapter 4] Assume that (47) holds. Then there exists λ * > 0 such that, for all 0 < λ ≤ λ * , (46) has a weak solution u ∈ W s,1 0 (Ω). Remark 0.4.3. a) The notion of weak solution to (46) will be given in Definition 4.1.2. b) The regularity results for (44) that we need to prove Theorem 0.4.5 are different from the ones used in Theorems 0.4.1 and 0.4.4. Nevertheless, the restriction s ∈ (1/2, 1) still arises out of these regularity results. See Proposition 4.3.5 for more details.

  ) : λc(x)w(x) = 0 a.e. x ∈ Ω , w = 1} . Using these notations, we state the following result which generalizes the results obtained in [18, Section 3]. Theorem 1.1.1. Assume that (A 0 ) holds and that λ ≤ 0. Then if m + p,λ > 0 and m - p,λ > 0, the problem (P λ ) has at least one solution. Remark 1.1.1.a) The space W λ depends only on the fact that λ = 0 or λ 0.

λFigure 1 . 1 :Figure 1 . 2 :

 1112 Figure 1.1: Illustration of Theorem 1.1.6 with h 0

Figure 1 . 3 :

 13 Figure 1.3: Existence regions of Theorem 1.1.7

Theorem 1 . 2 . 1 .

 121 [24, Theorems 3.1 and 4.2] Assume the existence of a non-decreasing function b :

  are respectively a lower and an upper solution to (1.3.1), then u 1 ≤ u 2 .

  ) : λc(x)w(x) = 0 a.e. x ∈ Ω , w = 1} and we define m := inf u∈W 1,p 0 (Ω)

  substituting in(1.7.4), we deduce thatΩ |∇φ 0 | p + (p -1) u | p-2 ∇e µ p-1 u ∇φ 0 dx = 0 .(1.7.5)Applying Proposition 1.2.4, this proves the existence of k ∈ R such thatφ 0 = ke µ p-1 u .As φ 0 = 0 and e µ p-1 u = 1 on ∂Ω, this implies that k = 0 which contradicts the fact that φ 0 > 0 in Ω.Proof of Theorem 1.1.3. The proof is just the combination of Proposition 1.7.1 and of Remark 1.1.2.Proof of Corollary 1.1.4. We see, combining Theorems 1.1.1 and 1.1.3, that if (P 0 ) has a solution then (P λ ) has a solution for any λ ≤ 0. Moreover this solution is unique by Theorem 1.1.2.

Step 3 :

 3 Conclusion.By Corollary 1.2.6, Proposition 1.2.7, and Lemma 1.5.2, we have the existence of v ∈ W ), local minimum of I λ and solution to (Q λ ) such that α λ ≤ v 0 as desired.

  ), u p * = 1 , and, for p = N , S N ,r = inf ∇u p p : u ∈ W

Remark 2 .

 2 1.1. a) Theorem 2.1.2, 1) generalizes [18, Theorem 1.2].

  with ϕ ≥ 0, it follows thatω |∇u| p-2 ∇u ∇ϕ dx + ω a(x)|u| p-2 u ϕ dx ≥ 0 .

Theorem 3 . 1 . 1 .

 311 Assume that (3.1.2) holds. Then (3.1.1) has a solution if, and only if, m d > 0. Remark 3.1.1. a) This result generalizes [18, Proposition 3.1 and Remark 3.2] and, for p = 2, [46, Theorem 1.3]. b) By [17, Theorem 1.1] we know that the solution obtained is unique.

Theorem 3 . 2 . 1 .

 321 [24, Theorems 3.1 and 4.2] Assume the existence of a non-decreasing function b :

.5. 3 )

 3 Gathering (3.5.2) and (3.5.3) we deduce that 0

- 1 .

 1 1 ∈ C 1 0 (Ω) strict upper solution to (P λ ). Let us then introduce By Lemma 3.4.5 and Remark 3.4.3, it follows that v 0 , v 1 ∈ C 1 0 (Ω) are a couple of well ordered strict lower and upper solutions to (Q λ ). Hence, applying Corollary 3.2.3 and Proposition 3.2.4 we have the existence of v ∈ C 1 0 (Ω) minimizer of I λ on M := v ∈ H 1 0 (Ω) : v 0 ≤ v ≤ v 1 , local minimum of I λ in the H 1 0 -topology and solution to (Q λ ). Moreover, by its construction v v 0 .

Lemma 3 . 7 . 1 .

 371 [START_REF] Kim | Green's function for second order elliptic equations with singular lower order coefficients[END_REF] Lemma 4.2] Let Ω ⊂ R N be a bounded domain, β ∈ (L N (Ω)) N and ξ ∈ L N /2 (Ω) with ξ ≥ 0 a.e. in Ω. Then, for every F ∈ H -1 (Ω), there exists an unique u ∈ H 1 0 (Ω) solution to-∆u + β(x), ∇u + ξ(x)u = F,

.7. 3 )Lemma 3 . 7 . 3 .

 3373 Then, we define u = w + ϕ and we observe that u ∈ H 1 (ω) is a solution to (3.7.2). Next, by[START_REF] Kim | Green's function for second order elliptic equations with singular lower order coefficients[END_REF] Proposition 3.10] we deduce that 0 ≤ u ≤ 1 in ω and, by [82, Theorem II-15.1] we obtain that u ∈ C 1,τ (ω) for some τ > 0. Finally, the uniqueness follow from[START_REF] Kim | Green's function for second order elliptic equations with singular lower order coefficients[END_REF] Proposition 3.10]. Let ω = B 1 (0) \ B 1/2 (0), ε ∈ (0, 1/4), x 0 ∈ ∂B 1 (0) and T : R N → R N given by T (x) = ε -1 (xx 0 ) + x 0 . Then, it follows that ω ⊂ T (ω) := {T (x) : x ∈ ω}.

.7. 6 )Lemma 3 . 7 . 4 .

 6374 Hence, the result follows from (3.7.4)-(3.7.6).Let ω = B 1 (0) \ B 1/2 (0), B = (B 1 , . . . , B N ) ∈ (L p (ω)) N and c ∈ L p (ω) for some p > N , ε ∈ [0, 1/4], B ε (y) = (B 1 ε , . . . , B N ε ) = εB(ε(yx 0 ) + x 0 ) and c ε (y) = ε 2 c(ε(yx 0 ) + x 0 ). Then, it follows that

Lemma 3 . 7 . 5 .

 375 Let ω = B 1 (0)\B 1/2 (0), B = (B 1 , . . . , B N ) ∈ (L p (ω)) N and c ∈ L p (ω)for some p > N . Then, there exists M > 0 such that, for all ε ∈ [0, 1/4], any solution to (P ε ) satisfies u C 1 (ω) ≤ M.

Corollary 3 . 7 . 7 .

 377 For z ∈ R N and R > 0, let β ∈ (L p (B R (z)) N and ξ ∈ L p (B R (z)) for some p > N such that ξ ≥ 0 a.e. in B R (z). Let x 0 ∈ ∂B R (z) and let u ∈ C 1 (B R (z)) be an upper solution to -∆u + β(x), ∇u + ξ(x)u = 0, in B R (z), u = 0, on ∂B R (z), (3.7.18) such that u(x) > u(x 0 ) = 0 for all x ∈ B R (z). Then ∂u ∂ν (x 0 ) < 0, where ν denotes the exterior unit normal. Proof. Let us define y = T (x) = 1 R (xz) and introduce the functions v(y) = u(Ry + z), B(y) = Rβ(Ry + z), c(y) = R 2 ξ(Ry + z). (3.7.19)

Definition 4 . 1 . 1 .

 411 We say that u is a weak solution to (P λ ) if u andD 2 s (u) belong to L 1 (Ω), u ≡ 0 in CΩ := R N \ Ω and Ω u(-∆) s φ dx = Ω µ(x)D 2 s (u) + λf (x) φ dx, ∀ φ ∈ X s , (4.1.6)whereX s := ξ ∈ C(R N ) : Supp ξ ⊂ Ω, (-∆) s ξ(x) exists ∀ x ∈ Ω and |(-∆) s ξ(x)| ≤ C for some C > 0 . (4.1.7)

Theorem 4 . 1 . 3 .

 413 Let Ω ⊂ R N , N ≥ 2, be a bounded domain with ∂Ω of class C 2 , let s ∈ (0, 1) and let µ ∈ L ∞ (Ω) such that µ(x) ≥ µ 1 > 0. Then, for all 1 ≤ p < N 2s , there exists f ∈ L p (Ω) such that (P λ ) has no weak solutions in W s,2 0 (Ω) for any λ > 0. Using the same kind of approach than in Theorem 4.1.1, i.e. regularity results for (4.1.8) and our fixed point argument, one can obtain existence results for related problems involving different nonlocal diffusion terms and different nonlinearities.First, we deal with the Dirichlet problem

Remark 4 . 1 . 5 .

 415 The regularity results for (4.1.8) that we need to prove Theorem 4.1.5 are different from the ones used in Theorems 4.1.1 and 4.1.4. Nevertheless, the restriction s ∈ (1/2, 1) still arises out of these regularity results. See Proposition 4.3.5 for more details.

  .1.8). Section 4.4 is devoted to the proofs of Theorems 4.1.1 and 4.1.4. Section 4.5 contains the proofs of Theorems 4.1.2 and 4.1.3. Section 4.6 deals with (Q λ ) and ( Q λ ), i.e., it is devoted to the proofs of Theorems 4.1.5 and 4.1.6.

Notation. 1 ) 2 ) 3 ) 4 )

 1234 In R N , we use the notations |x| = x 2 1 + . . . + x 2 N and B R (y) = {x ∈ R N : |x -y| < R}. For a bounded open set Ω ⊂ R N we denote its complementary as CΩ, i.e. CΩ = R N \ Ω. For p ∈ (1, ∞), we denote by p the conjugate exponent of p, namely p = p/(p-1) and by p * s the Sobolev critical exponent i.e. p * s = N p N -sp if sp < N and p * s = +∞ in case sp ≥ N . For u ∈ L ∞ (Ω) we use the notation u ∞ = u L ∞ (Ω) = esssup x∈Ω |u(x)|.

  u W s,p (Ω) := u p L p (Ω) + Ω×Ω |u(x)u(y)| p |x -y| N +sp dxdy 1 p .

1 ) [ 3 , Lemma 3 . 4 ]

 1334 be a bounded domain with boundary ∂Ω of class C 2 such that 0 ∈ Ω, 0 < s < 1 and p > 1. Then: If we set

Corollary 4 . 3 . 2 .

 432 Assume (4.3.2) and let v be the unique weak solution to (4.3.1):

Remark 4 . 3 . 3 .

 433 The proofs of Propositions 4.3.1, 4.3.3 and 4.3.4 are postponed to Subsection 4.3.1

Corollary 4 . 3 . 6 . 1 )

 4361 Assume (4.3.2) and let v be the unique weak solution to (4.3.1): If m = 1, then |∇ s v| ∈ L p (Ω) for all 1 ≤ p < N N -s and there exists C 4 = C 4 (s, p, Ω) > 0 such that

Lemma 4 . 3 . 8 .

 438 [START_REF]Towards a deterministic KPZ equation with fractional diffusion: the stationary problem[END_REF] Lemma 2.15] Assume (4.3.2) and let v be the unique weak solution to (4.3.1):1) If m = 1, then v ∈ W 1,p (R N ) for all 1 ≤ p < N N -(2s-1) and there exists C 6 = C 6 (s, p, Ω) > 0 such that

4. 3 . 1

 31 Proofs of Propositions 4.3.1, 4.3.3 and 4.3.4.

4. 3 . 2

 32 Proofs of Proposition 4.3.5 and Corollary 4.3.6 Next, using again Lemma 4.3.8 but with a different approach, we prove Proposition 4.3.5. As a consequence we will obtain Corollary 4.3.6. Proof of Proposition 4.3.5. Let v be the unique weak solution to (4.3.1) and define, for x ∈ R N arbitrary, S 1 := {y ∈ R N : dist(y, Ω) > 2} and S 2 := {y ∈ R N : dist(y, Ω) ≤ 2 and |x -y| ≥ 1}.

  have that|∇ s u(x)| ≤ C R N |∇u(y)| |x -y| N +s-1 dy. (4.3.30)

1 ∈

 1 s,p 0 (Ω) for all 1 ≤ p < N N -s .Proof. First observe that, g (t) = 0 if and only ift = t * :(0, ∞).Moreover, observe that g (t * ) = (p -1)pa p b 2 deduce that g has an strict global minimum on t = t * . Finally, observe that g(t * ) = a p 1 g(0) > 0 and lim t→∞ g(t) = ∞.

Lemma 4 . 4 . 2 .

 442 Let Ω ⊂ R N be a bounded domain with boundary ∂Ω of class C 0,1 and let s ∈ (0, 1). For all ε > 0 satisfying 0 < sε < s + ε < 1 and all 1 ≤ σ < r there existsC 9 = C 9 (s, ε, σ , r, Ω) > 0 such that R N |u(x)u(y)| σ |x -y| N +sσ dy 1 σ L r (Ω) ≤ C 9 u W s+ε,r 0 (Ω) , ∀ u ∈ W s+ε,r 0 (Ω) . (4.4.1)Proof. First of all, observe thatΩ R N |u(x)u(y)| σ |x -y| N +sσ dy r σ dx = Ω R N ∩{|x-y|<1} |u(x)u(y)| σ |x -y| N +sσ dy + R N ∩{|x-y|≥1}|u(x)u(y)| σ |x -y| N +sσ dy c r,σ (J 1 + J 2 ) .

( 4 . 4 . 2 )=J 2 ≤ C J 2 Ω

 44222 Let us then estimate J 1 . Applying Hölder inequality, we have that)u(y)| r |x -y| N +(s+ε)r dy R N ∩{|x-y|<1} dy |x -y| N -C J 1 (ε, σ , r) < ∞,we deduce thatJ 1 ≤ C J 1 Ω R N ∩{|x-y|<1} |u(x)u(y)| r |x -y| N +(s+ε)r dy dx ≤ C J 1 u r W s+ε,r 0 (Ω) . (4.4.3)Now, arguing as with J 1 , we obtain that )u(y)| r |x -y| N +(s-ε)r dyR N ∩{|x-y|≥1} dy |x -y| N + εσ r R N ∩{|x-y|≥1} |u(x)u(y)| r |x -y| N +(s-ε)r dy dx ≤ C J 2 u r W s-ε,r 0 (Ω) ≤ C J 2 u r W s+ε,r 0 (Ω) . (4.4.4)The result follows from (4.4.2), (4.4.3) and (4.4.4).

4. 4 . 1

 41 Proof of Theorem 4.1.1

C 1 , 5 ) 2 , 2 R( 2 Ω(I 1 •

 152221 given by Proposition 4.3.1, 3) applied with t = s + ε and p = r, C 10 := C2 9 |Ω| r-2m rm , where C 9 is the constant given by Lemma 4.4.2, andλ * := 1 4 f L m (Ω) C 2 1 C 10 µ ∞ .By the definition of λ * and Lemma 4.4.1, we know there exists and unique l ∈ (0, ∞) such thatC 1 (C 10 µ ∞ l + λ * f L m (Ω) ) = lHaving fixed the above constants, we introduceE := v ∈ W s,1 0 (Ω) : D Ω |u(x)u(y)| r |x -y| N +(s+ε)r dxdy ≤ l rFirst of all, using the notation ψ n = ϕ nϕ and the reverse triangle inequality, we obtain thatD 2 s (ϕ n ) -D 2 s (ϕ) L 1 (Ω) = Ω R N |ϕ n (x)ϕ n (y)| 2 -|ϕ(x)ϕ(y)| 2 |x -y| N +2s dy dx ≤ Ω R N |ϕ n (x)ϕ n (y)| + |ϕ(x)ϕ(y)| |ψ n (x)ψ n (y)| |x -y| N +2s dy dx n (x)ϕ n (y)| + |ϕ(x)ϕ(y)|Applying then Hölder inequality, we deduce thatD 2 s (ϕ n ) -D 2 s (ϕ) L 1 (Ω) ≤ Ω R N (|ϕ n (x)ϕ n (y)| + |ϕ(x)ϕ(y)|) 2 |x -y| N +2s dy 1 N |ψ n (x)ψ n (y)| 2 |x -y| N +2s dy |ϕ n (x)ϕ n (y)| + |ϕ(x)ϕ(y)|) 2 |x -y| N +2s dy dx 1 R N |ψ n (x)ψ n (y)| 2 |x -y| N +2s dy dx |ϕ n (x)ϕ n (y)| + |ϕ(x)ϕ(y)|) 2 |x -y| N +2s dy dx I 2 .

I 1 ≤ 2 Ω

 2 R N |ϕ n (x)ϕ n (y)| 2 |x -y| N +2s dy dx + Ω R N |ϕ(x)ϕ(y)| 2 |x -y| N +2s dy dx = 2 D 2 s (ϕ n ) L 1 (Ω) + D 2 s (ϕ) L 1 (Ω) .

  |ψ n (x)ψ n (y)|2 |x -y| N +2s dy dx= Ω R N |ψ n (x)ψ n (y)| θ |x -y| (N +s)θ |ψ n (x)ψ n (y)| 2-θ |x -y| N (1-θ)+s(2-θ) dy dx = ( * )|ψ n (x)ψ n (y)| |x -y| N +s dy θ

W. 2 . 4 . 4 . 8 .

 2448 Hölder and Sobolev inequalities and using that W s+ε,r 0 (Ω) ⊂ W s,r 0 (Ω) with continuous inclusion, we obtain thatϕ D 2 s (ϕ) m L m (Ω) ≤ S N ,r C ϕ mSince r > 2, the result follows from Lemma 4.4.Corollary Assume that (A 1 ) holds. Then T 1 is well defined.Proof. Since Ω is a bounded domain and m > N 2s > 1 the result follows from Lemma 4.4.7 arguing as in the proof of Lemma 4.4.3.Lemma 4.4.9. Assume (A 1 ) and let 0 < λ ≤ λ * . Then T 1 (E 1 ) ⊂ E 1 .Proof. Let us consider an arbitrary ϕ ∈ E and define u = T 1 (ϕ). By Corollary 4.3.4, since that 0 < λ ≤ λ * , we have thatD Ω |u(x)u(y)| r |x -y| N +(s+ε)r dxdy 1 r ≤ C 2 µ ∞ ϕ D 2 s (ϕ) L m (Ω) + C 2 λ * f L m (Ω) . (4.4.21) Hence, since ϕ ∈ E, by Lemma 4.4.7 and (4.4.18), it follows that D Ω |u(x)u(y)| r |x -y| N +(s+ε)r dxdy 1 r ≤ C 2 (C 11 µ ∞ l + λ * f L m (Ω) ) = l 1 3 .

ϕ n D 2 s 3 .

 23 (ϕ n ) → ϕ D 2 s (ϕ), in L 1 (Ω).(4.4.22) First observe that, since r > N s > N s+ε , for all ϕ ∈ E 1 , it follows thatϕ L ∞ (Ω) ≤ C D Ω |ϕ(x)ϕ(y)| r |x -y| N +(s+ε)r dxdy ≤ l r (4.4.23)Hence, since ϕ n → ϕ in W s,1 0 (Ω), by Vitali's Convergence Theorem we deduce that ϕ n → ϕ in L α (Ω) for all 1 ≤ α < ∞.Next, observe thatϕ n D 2 s (ϕ n ) -ϕD 2 s (ϕ) L 1 (Ω) = ϕ n (D 2 s (ϕ n ) -D 2 s (ϕ)) + D 2 s (ϕ)(ϕ nϕ) L 1 (Ω) ≤ ϕ n (D 2 s (ϕ n ) -D 2 s (ϕ)) L 1 (Ω) + D 2 s (ϕ)(ϕ nϕ) L 1 (Ω) ≤ ϕ n ∞ D 2 s (ϕ n ) -D 2 s (ϕ) L 1 (Ω) + D 2 s (ϕ) L m (Ω)ϕ nϕ L m (Ω) =: I 1 + I 2(4.4.24)

4. 5 1 Dµ 1 2 D 2 Dµ 1 4 D

 51224 Proofs of Theorems 4.1.2 and 4.1.3 In this section we prove Theorems 4.1.2 and 4.1.3. The aim of these theorems is to justify the hypotheses considered in Theorem 4.1.1. First we prove that (P λ ) has no solutions for λ large and so, that the smallness condition is somehow necessary to have existence of solution.Proof of Theorem 4.1.2. Assume that (P λ ) has a solution u ∈ W s,2 0 (Ω) and let φ ∈ C ∞ 0 (Ω) be an arbitrary function such thatΩ f (x)φ 2 (x)dx > 0,Considering φ 2 as test function in (P λ ) we observe thatΩ (-∆) s u φ 2 (x)dx = Ω µ(x) R N |u(x)u(y)| 2 |x -y| N +2s φ 2 (x)dydx + λ Ω f (x)φ 2 (x)dx. (4.5.1)Now, on one hand, since µ(x) ≥ µ 1 > 0 andD 2 s is symmetric in x, y, it follows that Ω µ(x) R N |u(x)u(y)| 2 |x -y| N +2s φ 2 (x)dydx = D Ω µ(x) |u(x)u(y)| 2 |x -y| N +2s φ 2 (x)dydx ≥ µ Ω |u(x)u(y)| 2 |x -y| N +2s φ 2 (x)dydx = Ω |u(x)u(y)| 2 |x -y| N +2s φ 2 (x)dydx +µ 1 Ω |u(x)u(y)| 2 |x -y| N +2s φ 2 (y)dydx ≥ Ω |u(x)u(y)| 2 |x -y| N +2s (φ(x) + φ(y)) 2 dydx.

( 4 . 5 . 2 ) 1 |x|

 4521 On the other hand, by Young's inequality, it follows that Ω (-∆) s u φ2 (x)dx = D Ω (u(x)u(y))(φ 2 (x)φ 2 (y)) |x -y| N +2s dydx = D Ω (u(x)u(y))(φ(x)φ(y))(φ(x) + φ(y)) |x -y| N +2s dydx ≤ D Ω |u(x)u(y)||φ(x) + φ(y)| )u(y)| 2 |x -y| N +2s (φ(x) + φ(y)) 2 dydx + 1 µ 1 D Ω |φ(x)φ(y)| 2 |x -y| N +2s dydx (4.5.3) Hence, substituting (4.5.2) and (4.5.3) to (4.5.1), we deduce that, if (P λ ) has a solution, then1 µ 1 D Ω |φ(x)φ(y)| 2 |x -y| N +2s dydx ≥ λ Ω f (x)φ 2 (x)dx, (4.5.4) which gives a contradiction for λ large enough. Now, we prove Theorem 4.1.3. This theorem shows that the regularity considered on f is almost optimal. Just the limit case f ∈ L N 2s (Ω) remains open. Proof of Theorem 4.1.3. Without loss of generality we choose a bounded domain Ω with boundary ∂Ω of class C 2 such that 0 ∈ Ω. Consider then f (x) =

1 µ 1 2

 12 D Ω |φ(x)φ(y)| 2 |x -y| N +2s dydx ≥ λ ε Ω f (x)φ 2 (x)dx = λ ε Ω φ

.5. 7 ) 2 |x|

 72 Nevertheless, since m < N 2s , we can choose ε > 0 small enough to ensure that N -ε m > 2s. In that case, by Proposition 4.2.3, 2), we have that inf)φ(y)| 2 |x -y| N +2s dydx Ω |φ(x)|which contradicts (4.5.7). Hence, the result follows.

4. 6

 6 Proofs of Theorems 4.1.5 and 4.1.6

  4.1, we know that there exists an unique l ∈ (0, ∞) such thatC 3 ( µ L ∞ (Ω) |Ω| r-qm mr l + λ * f L m (Ω) ) = l

Lemma 4 . 6 . 1 . 1 . 4 . 6 . 1 .Lemma 4 . 6 . 2 .

 4611461462 Assume (B 1 ) and let 0 < λ ≤ λ * . Then T 2 is well defined, T 2 (E 2 ) ⊂ E 2 and T 2 is compact. Proof. The proof of this lemma follows as in Lemmas 4.4.3, 4.4.4 and 4.4.6 using Proposition 4.3.5 instead of Proposition 4.3.Remark The only point in the proof of the previous lemma where we use 0 < λ ≤ λ * is to show that T 2 (E 2 ) ⊂ E 2 . The rest holds for every λ ∈ R. Assume that (B 1 ) holds. Then T 2 is continuous.

( 4 . 6 . 3 )

 463 Hence, if we show thatµ(x) |(-∆) s 2 ϕ n | q -|(-∆) s 2 ϕ| q → 0 , in L 1 (Ω), (4.6.4) the result follows from Proposition 4.3.13. Directly, since ϕ n , ϕ ∈ E 2 and µ ∈ L ∞ (Ω), applying the Mean Value Theorem and Hölder inequality, we deduce thatµ(x) |(-∆) s 2 ϕ n | q -|(-∆) C is a positive constant depending only on µ L ∞ (Ω), l, q and Ω. By (4.6.5), if we show thatΩ |(-∆) s 2 (ϕ nϕ)| q dx → 0, (4.6.6) the continuity of the operator follows from Proposition 4.3.13.

-

  ∆u = λc(|x|)u + µ(|x|)|∇u| 2 + h(|x|) , in B R (0) , u = 0 , on ∂B R (0) , (P R λ )

5. 2

 2 Boundary weak Harnack inequality: the optimal ε ε εIn Theorem 2.3.1, Chapter 2, we prove a new boundary weak Harnack inequality. More precisely, for 1 < p < ∞, we consider the boundary value problem-∆ p u + a(x)|u| p-2 u = 0 , u ∈ W 1,p 0 (Ω) . (P BHP )and we prove the following result. Theorem 5.2.1. [Theorem 2.3.1, Chapter 2] Let Ω ⊂ R N , N ≥ 2, be a bounded domain with boundary ∂Ω of class C 1,1 and let a ∈ L ∞ (Ω) be a non-negative function. Assume that u is a non-negative upper solution to (26) and let x 0 ∈ ∂Ω. Then, there exist R > 0, ε = ε(p, R, a ∞ , Ω) > 0 and C = C(p, R, ε, a ∞ , Ω) > 0 such that, for all R ∈ (0, R] , inf B R (x 0 )∩Ω u(x) d(x, ∂Ω) ≥ C B R (x 0 )∩Ω u(x) d(x, ∂Ω) ε dx 1/ε .

Theorem 5 . 2 . 2 .

 522 Let Ω ⊂ R N , N ≥ 2, be a bounded domain with boundary ∂Ω of class C 1,1 . Assume that u is a non-negative upper solution to (P BH-2 ) and let x 0 ∈ ∂Ω. Then, there exist R > 0 andC = C( A W 1,p (Ω) , λ, p, b L p (Ω) ) > 0 such that, for all R ∈ (0, R], inf B R (x 0 )∩Ω u(x) d(x, ∂Ω) ≥ C B R (x 0 )∩Ω u(x) d(x, ∂Ω) s dx 1/s f L p (B 2 R(x 0 )∩Ω) ,for each s < 1.

  with A A A and B B B satisfying suitable regularity and growth assumption on the line of the considered in[START_REF] Malý | Fine regularity of solutions of elliptic partial differential equations[END_REF] Chapter 3].

  )u(y)| 2 |x -y| N +2s dy .

1 )

 1 For p ∈ [1, +∞[, the norm ( Ω |u| p dx) 1/p in L p (Ω) is denoted by • p . We denote by p the conjugate exponent of p, namely p = p/(p -1) and by p * the Sobolev critical exponent i.e. p * =

	p	N p N -p if p < N and

  |∇e tu 2 | p-2 ∇e tu 2 ∇e tu 1 dx |∇e tu 1 | p-2 ∇e tu 1 ∇e tu 2 dx ≤ 0 .

				e tu 1 e tu 2	p	|∇e tu 2 | p -p	e tu 2 e tu 1	p-1
	+	{u 1 >u 2 }	t -µ t p e t(p-1)u 2	|∇e tu 2 | p + (p -1)	e tu 2 e tu 1	p	|∇e tu 1 | p -p	e tu 1 e tu 2	p-1

1)u 2 |∇e tu 2 | p + (p -1) e tu 2 e tu 1 p |∇e tu 1 | pp e tu 2 e tu 1 p-1 |∇e tu 1 | p-2 ∇e tu 1 ∇e tu 2 . Then, by (1.3.3), we have {u 1 >u 2 } tµ t p e t(p-1)u 1 |∇e tu 1 | p + (p -1) (1.3.4) By Picone's inequality (Proposition 1.2.4), we know that both brackets in (1.3.4) are positive and are equal to zero if and only if e tu 1 = ke tu 2 for some k ∈ R. As tµ > 0, thanks to (1.3.4), we deduce the existence of k ∈ R such that

  , Theorem IX-2.2]. Proof of Theorem 1.1.2. The proof is just the combination of Corollary 1.3.2 and Lemma 1.3.3. Remark 1.3.2. It is important to note that this comparison and uniqueness results do not hold in general for solution belonging only to W Remark 1.3.3. Under the assumption (1.3.2), a comparison principle for the problem

	1,p 0 (Ω). See [96, Example 1.1]

.

  Proposition 1.6.1. Assume that (A 1 ) holds, λ ≤ 0 and that m p,λ > 0. Then m is finite and it is reached by a function v ∈ W Consequently the problem (P λ ) has a solution.Proof. To prove that I λ has a global minimum since, by Lemma 1.5.3, any bounded Cerami sequence has a convergent subsequence it suffices to show that I λ is coercive. Having found a global minimum v ∈ W

	1,p 0 (Ω). 1,p 0 (Ω) we deduce, by Lemma 1.5.2, that u =	p-1 µ ln 1 +	µ p-1 v is a solution to (P λ ).
	To show that I λ is coercive we consider an arbitrary sequence {v n } ⊂ W	1,p 0 (Ω) such that v n → ∞
	and we prove that
	lim n→∞

  1.8.2 holds. By Proposition 1.8.4 we have a first critical point, which is a local minimum of I

λ . On the other hand, since the Cerami condition holds, in view of Lemmata 1.8.2. and 1.8.3, we can apply Theorem 1.2.8 and obtain a second critical point of I λ at the mountain-pass level. This gives two different solutions to (Q λ ). Finally, by Lemma 1.5.2, we obtain two solutions to (P λ ).

  are both upper solutions to (P λ,k ) with β 1 β 2 0. By Proposition 1.4.2, (P λ,k ) has a lower solution α with α ≤ β 1 and hence, by Theorem 1.2.1, the problem (P λ,k ) has a minimum solution u 1

  Hence v is an upper solution to (2.3.8). Case 1: ρ ≥ ρ /4. Observe that v/M j is a non-negative upper solution to (2.3.8). Moreover, as ρ ≤ ρ , (2.3.13) implies that

	Now, we distinguish two cases:

  1 0 (Ω) is equipped with the norm u := Ω |∇u| 2 dx For p ∈ [1, +∞[, the norm ( Ω |u| p dx) 1/p in L p (Ω) is denoted by • p . We denote by p the conjugate exponent of p and by 2 * the Sobolev critical exponent i.e. 2 * = 2N /(N -2) if N ≥ 3 and 2

	1/2	.
	6)

  .1.8) In Section 4.3, we obtain sharp Calder ón-Zygmund type regularity results for the fractional Poisson equation (4.1.8) with low integrability data. We believe these results are of independent interest and will be useful in other settings. Actually, Section 4.3 can be read as an independent part of the present chapter. In particular, we refer the interested reader to Propositions 4.3.1, 4.3.3 and 4.3.4.

  ) . Assume (4.3.2) and let v be the unique weak solution to (4.3.1) and t ∈ (s, 1).If ≤ v W t,p (R N ) ≤ C 2 h L m (Ω) . Remark 4.3.2.Notice that, in the case where t ∈ (s, 1), Propositions 4.3.1 and 4.3.4 complete and somehow give a more precise information than the result obtained in[START_REF]Nonlocal self-improving properties[END_REF].

	Proposition 4.3.4. N 2s ≤ m < N s then v ∈ W 0 (Ω) for all 1 ≤ p < t,p	mN N -m(2s-t) and there exists C 2 = C 2 (m, s, t, p, Ω) > 0 such
	that
	v W t,p 0 (Ω)

  for all 1 ≤ p < ∞ and there existsC 4 = C 4 (s, m, p, Ω) > 0 such that ∇ s v L p (Ω) ≤ C 4 h L m (Ω) .Remark 4.3.4. The proofs of Proposition 4.3.5 and Corollary 4.3.6 will be given in Subsection 4.3.2 Now, before proving Propositions 4.3.1, 4.3.3, 4.3.4, 4.3.5 and Corollary 4.3.6, we state some known results that will be useful in our proofs. First of all, we gather in the following lemma several results of[START_REF] Leonori | Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations[END_REF].

	Lemma 4.3.7. [84,

  Lemma 4.3.9. [28,Theorem 1] Assume that (4.3.3) holds and (4.3.4) fails. Then, for every θ ∈ (0, 1), there exists a constant C = C(s 1 , s 2 , p 1 , p 2 , θ) > 0 such that

				.3.3)
	Moreover, let us introduce the condition
	s 2 = p 2 = 1 and	1 p 1	≤ s 1 .	(4.3.4)

  Let v be the unique weak solution to(4.3.1). By Lemma 4.3.7, 3), we know that (4.3.10) holds. On the other hand, by Lemma 4.3.8, 3), we have thatv W 1,p 2 (R N ) ≤ C 6 h L m (Ω) , ∀ 1 ≤ p 2 < ∞. (4.3.11)The result follows from Lemma 4.3.9 applied with η = t, s 1 = 0 and s 2 = 1.Remark 4.3.5. The restriction s ∈ (1/2, 1) comes from Lemma 4.3.8. It is not expected that Lemma 4.3.8 holds true for s ∈ (0, 1/2]. Hence, with this approach we are limited to deal with s ∈ (1/2, 1). Now, using the Bessel potential space L s,p (R N ) (see Definition 4.2.3), Theorem 4.2.4 and Lemma 4.3.10 below, we prove Propositions 4.3.3 and 4.3.4. We begin proving a regularity result in the Bessel potential space. Proposition 4.3.11. Assume (4.3.2) and let v be the unique weak solution to (4.3.1):

	4) Lemma 4.3.10. [9, Theorem 7.58] Let 0 < t < s < 1, 1 < p < N s-t and q =	N p N -p(s-t) . Then W s,p (R N ) ⊂
	W t,q (R N ) with continuous inclusion.
										.3.10)
	By Lemma 4.3.8, 2), we know that (4.3.9) holds. Moreover, by Lemma 4.3.9 applied with η = t,
	s 1 = 0 and s 2 = 1, it follows that (4.3.7) holds. The result follows from (4.3.7), (4.3.9) and (4.3.10)
	using that
	1 ≥	1 p	=	1 -t p 1	+	t p 2	>	t(N -m(2s -1)) mN	.

  Let v be the unique weak solution to (4.3.1). In this case, by Proposition 4.3.11, we know that v ∈ L s,p (R N ) for all 1 ≤ p < ∞. Hence, by Theorem 4.2.4, we conclude.Having at hand Lemmas 4.3.10 and 4.3.12, we prove Propositions 4.3.3 and 4.3.4.

	1) N 2s < m < N
	s ,p 0 (Ω) for all 0 < s < s and
	v W s ,p 0 (Ω) ≤ C v L s,p (R 2) m ≥ N s
	Proof of Proposition 4.3.3. First observe that, since t ∈ (0, s) it follows that N 2s-t < N s . Then we split
	into two cases:
	1

s

Let v be the unique weak solution to (4.3.1). By Proposition 4.3.11, 2) we know that v ∈ L s,p (R N ) for all 1 ≤ p ≤ mN N -ms . Thus, by Theorem 4.2.4 we conclude that v ∈ W N ) ≤ C 8 h L m (Ω) .

  Proof ofCorollary 4.3.6. By [95, Lemma 15.9] we know that

	.3.28)
	Taking into account (4.3.26)-(4.3.28), the result follows from (4.3.23).
	3) m ≥ N 2s-t .
	The result follows arguing as in the proof of Proposition 4.3.11, 3) using Proposition 4.3.5,
	2).

Je suis en train d'écrire les remerciements de ma thèse donc la première chose à faire est clairement de remercier à mes parents-de-thèse Colette DE COSTER et Louis JEANJEAN. Je n'arrive pas à imaginer deux meilleures personnes pour m'avoir dirigé ces trois ans. MERCI de m'avoir guidé pendant ces trois ans, MERCI pour toutes les incroyables opportunités mais, surtout, MERCI de m'avoir donné toute la confiance du monde depuis le premier moment. C ¸a a été (et j'espère c ¸a sera) un plaisir de travailler avec vous.

MERCI Colette pour ton amabilité infinie, MERCI pour ton amour des maths et ton envie constante d'aller plus loin, d'apprendre et d'enseigner mais, surtout, MERCI de m'avoir enseigné à avoir de la patience (work in progress).

MERCI Louis pour ton incroyable disponibilité et ton efficacité absolue, MERCI pour toute la liberté que tu m'as donné et pour la clarté que tu as pour comprendre la recherche mais, surtout, MERCI de m'avoir enseigné comment écrire des maths (work in progress).

1,p 0 (Ω) and m + p,λ > 0 and m - p,λ > 0 holds for example under a suitable smallness condition on µ + h + and µ -h -. In particular observe that, if µ and h have opposite signs, then m + p,λ > 0 and m - p,λ > 0.

c ómo comenzar mi tesis, he llegado a la siguiente conclusi ón: No hay mejor manera de comenzar que diciendo ¡GRACIAS! GRACIAS a todas aquellas personas que, a lo largo de estos tres extra ños a la par que magníficos a ños, han perdido un instante de su tiempo para compartirlo conmigo.

Siendo esto unos agradecimientos de tesis creo que está claro quién debe encabezar estas líneas. Un gran GRACIAS a mis padres-de-tesis 1 Colette DE COSTER y Louis JEANJEAN. No consigo imaginar dos personas mejores que vosotros para haberme guiado durante estos tres a ños. GRA-CIAS por confiar en mí, GRACIAS me habéis dado, pero, sobre todo, GRACIAS por ponérmelo todo tan "fácil" desde a ños con vosotros.

GRACIAS Colette 2 por tu amabilidad infinita (ni una mala cara te recuerdo), GRACIAS más lejos, de aprender y ense ñar pero, sobre todo, GRACIAS por ense ñarme a tener paciencia (work in progress).

GRACIAS Louis 3 por tu increíble disponibilidad y eficiencia absoluta, GRACIAS por la libertad concedida y la claridad que tienes para entender la investigaci ón pero, sobre todo, GRACIAS por ense ñarme c ómo escribir matemáticas (work in progress).

Una vez dicho esto, GRACIAS Tommaso. GRACIAS por los quinientos Skype de estos a ños y las invitaciones a Roma, pero, sobre todo, GRACIAS por apostar por mí allá por el final de 2015. De igual manera, GRACIAS David. Muchísimas GRACIAS por la facilidad que me has puesto siempre que te he pedido algo. A ún me cuesta tutearte (work in progress). Por otro lado, pero no por ello menos importante, no puedo olvidarme de Boumediene. Muchas GRACIAS por tu increíble amor a las matemáticas, innumerables ideas. Ha sido, , un placer trabajar contigo.

Remark 2.2.1. With the regularity imposed in the following lemmas and in the absence of a gradient term in the equation, we do not need the lower and upper solutions to be bounded. The full Definition 2.2.1 will however be needed in other parts of the chapter. Lemma 2.2.1. (Local Maximum Principle) Under the assumption (2.2.3), assume that u ∈ H 1 (ω) is a lower solution to (2.2.2). For any ball B 2R (y) ⊂ ω and any s > 0, there exists C = C(s, r, a L r (B 2R (y)) , R) > 0 such that sup B R (y)

Proof. See for instance [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Theorem 8.17] [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] for a ∈ L ∞ (ω) and s > 1. Nevertheless, as it is remarked on page 193 of that book, the proof is valid for a ∈ L r (ω) with r > N /2 and, following closely the proof of [START_REF] Malý | Fine regularity of solutions of elliptic partial differential equations[END_REF]Corollary 3.10], the proofs can be extended for any s > 0. Proof. See for instance [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Theorem 8.18] and [START_REF] Malý | Fine regularity of solutions of elliptic partial differential equations[END_REF]Theorem 3.13].

Now, inspired by [START_REF] Brezis | Some simple nonlinear PDE's without solutions[END_REF]Lemma 3.2] (see also [60, Appendix A] ), we establish the following version of the Brezis-Cabré Lemma. Lemma 2.2.4. Let ω ⊂ R N , N ≥ 2, be a bounded domain with boundary ∂ω of class C 1,1 and let a ∈ L ∞ (ω) and f ∈ L 1 (ω) be non-negative functions. Assume that u ∈ H 1 (ω) is an upper solution to -∆u + a(x)u = f (x) , u ∈ H 1 0 (ω) . Then, for every B 2R (y) ⊂ ω, there exists C = C(R, y, ω, a ∞ ) > 0 such that

Proof. First of all, as a and f are non-negative, by the weak maximum principle, it follows that inf ω u(x) d(x, ∂ω) ≥ 0 . Now let B 2R (y) ⊂ ω . By the above inequality, we can assume without loss of generality that

We split the proof into three steps.

A priori bound

This section is devoted to the proof of Theorem 2.1.1. As a first step we observe that, to obtain our a priori upper bound on the solutions to (P λ ), we only need to control the solutions on Ω + . This can be proved under a weaker assumption than (A 1 ). More precisely, we assume

and we prove the next result.

Lemma 2.4.1. Assume that (B) holds. Then, there exists M > 0 such that, for any λ ∈ R, any solution u of (P λ ) satisfies sup

Remark 2.4.1. Let us point out that if c + ≡ 0, i.e. |Ω + | = 0, the problem (P λ ) reduces to

which is independent of λ. If (2.4.1) has a solution, by [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF]Proposition 4.1] it is unique and so, we have an a priori bound.

Proof. In case problem (P λ ) has no solution for any λ ∈ R, there is nothing to prove. Hence, we assume the existence of λ ∈ R such that (P λ) has a solution ũ. We shall prove the result with M := 2 ũ ∞ . Let u be an arbitrary solution to (P λ ).

Step 1: u ≤ sup Ω + u + + M.

We then obtain

As v ≤ 0 on ∂D, the function v is a lower solution to

Setting ṽ = ũ + ũ ∞ we observe that

and thus, as ṽ ≥ 0 on ∂D, the function ṽ is an upper solution to (2.4.2). By [17, Lemma 2.1], we know that u, ũ ∈ H 1 (Ω)∩W 1,N loc (Ω)∩C(Ω) and hence, v, ṽ ∈ H 1 (D)∩W

Next, considering stronger regularity assumptions on the coefficient functions and combining lower and upper solution and variational techniques, we improve the conclusions of Theorem 3.1.3. We derive a more precise information on the structure of the set of solutions to (P λ ) when λ > 0. Let us first introduce the following order notions.

• u v if u < v and, for all x ∈ ∂Ω, either u(x) < v(x), or, u(x) = v(x) and ∂u ∂ν (x) > ∂v ∂ν (x), where ν denotes the exterior unit normal.

Under the assumption

we prove the following theorems. Theorem 3.1.4. Assume (A 2 ) and suppose (P 0 ) has a solution u 0 with c + u 0 0. Then, every u solution to (P λ ) with λ > 0 and c + u ≥ 0 satisfies u u 0 . Moreover, there exists λ ∈]0, +∞[, such that:

• for every λ ∈]0, λ[ , (P λ ) has at least two solutions u λ,1 , u λ,2 ∈ C 1 0 (Ω) such that u λ,1 u 0 ;

• (P λ ) with λ = λ has at least one solution u λ ∈ C 1 0 (Ω) such that u λ ≥ u 0 ;

• for λ > λ the problem (P λ ) has no solution u such that c + u ≥ 0.

Theorem 3.1.5. Assume (A 2 ) and suppose (P 0 ) has a solution u 0 with c + u 0 0. Then, for every λ > 0, the problem (P λ ) has at least two solutions u λ,1 , u λ,2 ∈ C 1 0 (Ω) such that u λ,1 u 0 . b) At the expense of considering µ > 0 constant instead of µ ∈ L ∞ (Ω) with µ(x) ≥ µ 1 > 0 in Ω, Theorems 3.1.4 and 3.1.5 extend the main existence results of [START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF] to the case where c may change sign. Moreover, unlike [START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF], we do not assume global sign conditions on u 0 (solution to (P 0 )). Hence, even in the case where c -≡ 0, i.e. c has a sign, our hypotheses are weaker than the corresponding ones in [START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF].

c) Theorem 3.1.4 removes the "thick zero set" condition on the support of c λ considered in [48, Theorem 1.2] and gives somehow a more precise information. In turn, here µ is constant and we require a stronger regularity on the coefficient functions c λ and h + . and

When λ is positive this functional becomes unbounded from below and presents a concaveconvex type geometry. Then, in trying to obtain critical points, we have to overcome several difficulties. First, we shall notice that g is only slightly superlinear at infinity. Hence, I λ does not satisfies an Ambrosetti-Rabinowitz type condition. Moreover, the coefficient functions c λ and h have no sign. In this context, to prove that the Palais-Smale sequences are bounded may be challenging. Our proof is inspired by [START_REF] Jeanjean | Multiple solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF]. However, since we do not impose h 0, the proof becomes more involved. The role of the lower solution that we will construct in Proposition 3.4.2 is again crucial. See Section 3.5 for more details.

Having at hand the Palais-Smale condition for I λ with λ > 0, we shall look for critical points which are either local minimum or of a mountain-pass type. In Theorem 3.1.3, we work mainly with variational techniques as in [START_REF] Jeanjean | Multiple solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF][START_REF] Jeanjean | Existence and multiplicity for elliptic problems with quadratic growth in the gradient[END_REF]. Nevertheless, since our hypotheses are weaker than the corresponding ones in [START_REF] Jeanjean | Multiple solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF][START_REF] Jeanjean | Existence and multiplicity for elliptic problems with quadratic growth in the gradient[END_REF], to prove that the mountain-pass geometry holds becomes much more involved. In Theorems 3.1.4 and 3.1.5 we combine lower and upper solution with variational techniques. In both theorems a first solution is obtained throughout the existence of well-ordered lower and upper solution. This solution is further proved to be a local minimum. Then, we obtain a second solution by a mountain-pass type argument.

Another key ingredient in the proofs of Theorems 3.1.4 and 3.1.5 is the following estimate that can be seen as a combination of the Strong maximum principle and the Hopf's Lemma with unbounded lower order coefficients. Under the assumption

Then, either u ≡ 0 or u 0. The rest of the chapter is organized as follows. In Section 3.2 we recall some auxiliary results that will be useful. Section 3.3 is devoted to the proof of Theorem 3.1.1. In Section 3.4 we construct the lower solution α λ and we present the functional setting to deal with (Q λ ). Section 3.5 is devoted to prove the Palais-Smale condition and to show that, if (P 0 ) has a solution, then I λ has a mountain-pass geometry. This allows us to prove Theorem 3.1.3. Section 3.6 is devoted to the proofs of Theorems 3.1.4 and 3.1.5 and Corollary 3.1.6. Finally, in Appendix 3.7, we prove the Hopf's Lemma with unbounded lower order terms. This permits to prove Theorem 3.1.7.

and

we can prove that

Proof of Theorem 3.

Now, by Young's inequality, observe that

Hence, gathering (3.3.3)-(3.3.4) and using the density of C ∞ 0 (Ω) in H 1 0 (Ω), we have that

and we know that W d ⊆ W 0 , we obtain inf

Assume by contradiction that

Then, by standard arguments there exists

and so, that φ 0 is an eigenfunction associated to the first eigenvalue (which we are assuming equal to zero) of the eigenvalue problem

Applying then [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Theorem 8.20] and arguing as in [39, Proposition 3.2] we deduce that φ 0 > 0 in Ω. Since d 0, this contradicts that φ 0 ∈ W d and the result follows.

The lower solution and the functional setting

The aim of this section is to introduce a variational problem which is completely equivalent to (P λ ). As explained in the introduction, the key to find this equivalent problem is the construction of a lower solution below every upper solution to (P λ ).

The construction of this lower solution relies on the following a priori lower bound proved by the first author and L. Jeanjean in [START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF]. Let us consider the boundary value problem

under the assumption

d and f belong to L q (Ω) for some q > N /2 , µ > 0. Proposition 3.4.2. Under the assumption (A 1 ), for any λ ∈ R, there exists u λ ∈ H 1 (Ω) ∩ L ∞ (Ω) lower solution to (P λ ) such that, for every β upper solution to (P λ ), we have u λ ≤ min{0, β}.

Proof. We shall consider separately the cases λ ≤ 0 and λ > 0. The case λ ≤ 0 can be obtained exactly as in [START_REF] Coster | Existence and multiplicity for elliptic p-Laplacian problems with critical growth in the gradient[END_REF]Proposition 4.2]. We turn then to study the case where λ > 0. Fixed λ > 0 arbitrary, let us denote by M λ,1 > 0 the constant given by Lemma 3.4.1 applied to (P λ ) and let us introduce the auxiliary problem

Thanks to Lemma 3.4.1, there exists M λ,2 > 0 such that, for every

, we introduce the problem

and denote by α k its solution. Since -λkc + (x)h -(x) -λM λ,1 c + (x) -1 < 0, the weak maximum principle implies that α k ≤ 0. Observe that, for every β 1 upper solution to (3.4.3), we have that

Consequently, it follows that -∆β 1 ≥ -∆α k , in Ω,

and, by the comparison principle, that β 1 ≥ α k . Now, we introduce the problem

where

Observe that β 1 and 0 are upper solutions to (3.4.5). Recalling that the minimum of two upper solutions is an upper solution, it follows that β = min{0, β 1 } is an upper solution to (3.4.5). As α k is a lower solution to (3.4.5) with α k ≤ β, applying Theorem 3.2.1, we conclude the existence of v λ,k minimal solution to (3.4.5) with

Now, observe that v λ,k is an upper solution to (3.4.3). Hence, it follows that v λ,k ≥ -M λ,2 > -k and so, that v λ,k is a solution to (3.4.3).

Finally, let us introduce u λ = v λ,k -M λ,1 and observe that

Hence, we have that u λ is a lower solution to (P λ ) with u λ ≤ -M λ,1 . Thus, since every β upper solution to (P λ ) satisfies β ≥ -M λ,1 , we have that u λ is a lower solution to (P λ ) with u λ ≤ β for every β upper solution to (P λ ).

Step 2: For every λ ∈]0, λ[, (P λ ) has at least two solutions u λ,1 , u λ,2 ∈ C 1 0 (Ω) such that u 0 u λ,1 . By Proposition 3.6.3, for λ < λ, there exists a first critical point v λ,1 ∈ C 1 0 (Ω), which is a local minimum of I λ . Since the Palais-Smale condition at any level d ∈ R holds, by Theorem 3.2.6, we have two options. If we are in the first case, then together with Lemma 3.5.5, we see that I λ has the mountain-pass geometry and by Theorem 3.2.5, we have the existence of a second solution to (Q λ ). In the second case, we have directly the existence of a solution to (Q λ ). Then, by Lemma 3.4.5, we conclude the existence of two solutions to (P λ ). By the construction of v λ,1 , it follows that u λ,1 = 1 µ ln(1 + µv λ,1 ) u 0 .

Step 3: Existence of solution to (P λ )

Let v λ the solution to (Q λ ) obtained in Proposition 3.6.3. Applying Lemma 3.4.5 we conclude that u λ = 1 µ ln(1 + µv λ ) is a solution to (P λ ) for λ = λ. Moreover, by construction u λ ≥ u 0 .

Step 4: For λ > λ the problem (P λ ) has no solutions u with c + u ≥ 0.

This follows directly from Lemma 3.6.2.

Proof of the first part of Corollary 3.1.6. Let u 0 be the unique solution to (P 0 ). Since (A 2 ) holds, we know that u 0 ∈ C 1 0 (Ω). Now, observe that

Hence, since h 0, by Theorem 3.1.7, it follows that that u 0 0 , and so, in particular that c + u 0 0. The corollary then follows immediately from Theorem 3.1.4. Now, we turn to prove Theorem 3.1.5 and the second part of Corollary 3.1.6.

Proposition 3.6.4. Assume (A 2 ) and suppose that (P 0 ) has a solution u 0 with c + u 0 0. For every λ > 0 there exists v ∈ C 1 0 (Ω) with v v 0 := 1 µ (e µu 0 -1), which is a local minimum of I λ in the H 1 0 -topology and a solution to (Q λ ).

Proof. By Corollary 3.4.3 we have the existence of a strict lower solution α λ with α λ β for every β upper solution to (P λ ). On the other hand, arguing as in Lemma 3.6.2 we prove that u 0 is a strict upper solution to (P λ ) for all λ > 0. Arguing exactly as in Proposition 3.6.3 we deduce the existence of v ∈ C 1 0 (Ω) local minimum of I λ and solution to (Q λ ). Moreover, by its construction v v 0 = 1 µ e µu 0 -1 .

Proof of Theorem 3.1.5. The result follows from Proposition 3.6.4 arguing exactly as in the proof of Theorem 3.1.4.

Proof of the second part of Corollary 3.1.6. Since h 0 and (A 2 ) holds, the problem (P 0 ) has always a solution u 0 ∈ C 1 0 (Ω). Moreover, observe that 0 is an upper solution to (P 0 ). Hence, by Lemma 3.6.1, it follows that u 0 ≤ 0. Now, as u 0 satisfies

we deduce by Theorem 3.1.7 that u 0 0 and, in particular that c + u 0 0. Thus, the corollary follows immediately from Theorem 3.1.5.

We consider then v

and we observe that v n solves

, on ∂B 1/2 (0).

(3.7.8)

Now, for all n ∈ N, let us define

and observe that w n solves

, in ω, w n = 0, on ∂ω.

(3.7.9)

By [66, Lemma 9.17], there exists C(ω, N ) > 0 such that

and so, since v n C 1 (ω) = 1 for all n ∈ N, by Lemma 3.7.4, there exists

From the above inequality, we deduce the existence of C 2 > 0 (independent of n) such that

Since p > N , by the Sobolev compact embedding, we deduce that v n → v in C 1 (ω) for some v ∈ C 1 (ω). This implies that v is a weak solution to

for some B ∈ (L p (ω)) N and c ∈ L p (ω). Hence, by [79, Proposition 3.10], we deduce that v ≡ 0. This contradict the fact that v n → v in C 1 (ω) and the result follows.

Having at hand all the needed ingredients, we prove the Hopf's Lemma with unbounded lower order terms. Lemma 3.7.6. (Hopf's Lemma) Let B ∈ (L p (B 1 (0)) N and c ∈ L p (B 1 (0)) for some p > N such that c ≥ 0 a.e. in B 1 (0). Let x 0 ∈ ∂B 1 (0) and let u ∈ C 1 (B 1 (0)) be an upper solution to

such that u(x) > u(x 0 ) = 0 for all x ∈ B 1 (0). Then ∂u ∂ν (x 0 ) < 0, where ν denotes the exterior unit normal.

Proof. Let us fix ω := B 1 (0) \ B 1/2 (0) and split the proof into several steps:

Proof. First of all observe that the existence of v n and v are insured by Lemma 4.3.7. Now, let us define w n = v nv and observe that w n satisfies

in Ω,

Applying Proposition 4.3.1 with m = 1, it follows that

and let h ∈ L 1 (Ω). Then the operator S :

. Now, applying Lemma 4.3.9 with η = s, s 1 = 0 and s 2 = 1, we obtain that This section is devoted to prove Theorems 4.1.1 and 4.1.4. As indicated in the introduction, once we have the regularity results of Section 4.3, we follow the approach first develop in [START_REF] Phuc | Morrey global bounds and quasilinear Riccati type equations below the natural exponent[END_REF]Section 6]. Let us begin with two elementary technical lemmas that will be useful in the proofs of both theorems. which is a closed convex set of W s,1 0 (Ω). Then, we define T : E → W s,1 0 (Ω) by T (ϕ) = u, where u is the weak solution to

and observe that problem (P λ ) is equivalent to the fixed point problem u = T (u). Hence, to prove Theorem 4.1.1, we shall show that T has fixed point belonging to W s,2 0 (Ω)∩C 0,α (Ω) for some α > 0. Lemma 4.4.3. Assume that (A 1 ) holds. Then T is well defined.

Proof. First of all, by Hölder inequality and Lemma 4.4.2, observe that for all ϕ ∈ E,

Hence, for all ϕ ∈ E, it follows that Proof. For an arbitrary ϕ ∈ E, we define u = T (ϕ). Now, by Proposition 4.3.1 and since 0 < λ ≤ λ * , it follows that

(4.4.9) Also, by Lemma 4.4.2, Hölder inequality and the definition of C 10 , we obtain that

Thus, since ϕ ∈ E, we have that

Hence, since by Proposition 4.3.1 we also know that u ∈ W s,1 0 (Ω), we conclude that u ∈ E and so, that T (E) ⊂ E. Lemma 4.4.5. Assume that (A 1 ) holds. Then T is continuous.

Proof. Let {ϕ n } ⊂ E be a sequence such that ϕ n → ϕ in W s,1 0 (Ω) and define u n = T (ϕ n ), for all n ∈ N, and u = T (ϕ). To show that u n → u in W s,1 0 (Ω), and so, that T is continuous, we prove that we know that T is continuous, compact and satisfies T (E) ⊂ E, we can apply the Schauder fixed point Theorem to obtain u ∈ E such that T (u) = u. Thus, we conclude that (P λ ) has a weak solution for all 0 < λ ≤ λ * . Finally, since u ∈ W s,1 0 (Ω) ∩ W s,r 0 (Ω) for some 1 < 2 < r, by Lemma 4.3.9 applied with s 1 = s 2 = s, we deduce that u ∈ W s,2 0 (Ω). Moreover, since r > N /s, by [55, Theorem 8.2], we know that every ϕ ∈ E belongs to C 0,α (Ω) for some α > 0. First observe that, as before, without loss of generality we can assume m ∈ ( N 2s , N s ), µ ∞ 0 and f L m (Ω) 0. Next, let us fix some notation. We fix r = 3mN N + ms (4.4.17)

and ε := ε(r, m, s) > 0 such that

Also, we introduce and fix the constants C 2 , given by Corollary 4.3.4 applied with t = s + ε and p = r, C 11 := S N ,r CC 2m 9 , where S N ,r is the optimal constant in the Sobolev inequality (Theorem 4.2.1), C is the smallest constant guaranteeing the continuous embedding W s+ε,r 0 (Ω) ⊂ W s,r 0 (Ω) and C 9 is the constant given by Lemma 4.4.2, and

.

Then, by Lemma 4.4.1 we know that there exists and unique l ∈ (0, ∞) such that

Having fixed all these constants, we define

, which is a closed convex set of W s,1 0 (Ω), and Observe that ( P λ ) is equivalent to the fixed point problem u = T 1 (u). Hence, we shall prove that T 1 has a fixed point belonging to W s,2 0 (Ω) ∩ C 0,α (Ω) for some α > 0. with λ a real parameter. Here Ω ⊂ R N , N ≥ 2, is a bounded domain with smooth boundary, c and h belong to L q (Ω) for some q > N /2 and µ belongs to L ∞ (Ω).

Our work in this direction is mainly devoted to the study of the non-coercive cases where λc 0 a.e. in Ω, i.e. λc 0 or λc change sign. Nevertheless, despite the results obtained and the significant number of recent papers by other authors [START_REF]Continuum of solutions for an elliptic problem with critical growth in the gradient[END_REF][START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF][START_REF] Jeanjean | Multiple solutions for an indefinite elliptic problem with critical growth in the gradient[END_REF][START_REF] Jeanjean | Existence and multiplicity for elliptic problems with quadratic growth in the gradient[END_REF][START_REF] Nornberg | A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient[END_REF][START_REF] Souplet | A priori estimates and bifurcation of solutions for an elliptic equation with semidefinite critical growth in the gradient[END_REF], the structure of the set of solutions is far from being well understood. So far, concerning multiplicity results, just the existence of two solutions (under suitable hypotheses) has been established. The main question we would like to address in this direction is the existence of more than two solutions for c 0 and λ > 0 large enough.

The phase plane analysis obtained in [START_REF] Coster | Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient[END_REF]Section 6] for N = 1 and µ, c and h positive constants seems to indicate that, for λ > 0 large enough, the problem (P λ ) has more than two solutions. Furthermore, it seems that there exists a link with the spectrum of the boundary value problem -∆u = γc(x)u, u ∈ H 1 0 (Ω).

For the particular case µ constant and h ≡ 0 we hope to obtain our first higher multiplicity results adapting the lower and upper solutions techniques introduced by C. De Coster in [START_REF] Coster | Sign-changing solutions and multiplicity results for elliptic problems via lower and upper solutions[END_REF]. More precisely, for µ constant, h ≡ 0 and λ > 0 large enough we hope to obtain three non-trivial solutions and the trivial one. Nevertheless, as soon as h 0, we cannot use this approach and new ideas are needed. In particular, we wish to determine which assumptions on h give rise to more than two solutions to (P λ ). This is certainly a challenging open problem, but we believe that it