Salvador López-Martínez
email: salvador.lopez-martinez@inria.fr

A BLOW-UP APPROACH FOR SINGULAR ELLIPTIC PROBLEMS WITH NATURAL GROWTH IN THE GRADIENT

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Introduction

Let Ω ⊂ R N (N ≥ 3) be a bounded domain of class C 2 , g : Ω×(0, +∞) → R be a Carathéodory function, and f : [0, +∞) → [0, +∞) be a continuous function. In this work we will study the existence of solution to elliptic problems of the following form:

(P λ )      -∆u + g(x, u)|∇u| 2 = λf (u), x ∈ Ω, u > 0, x ∈ Ω, u = 0, x ∈ ∂Ω,
where λ > 0 is a parameter. The precise conditions on functions g, f and the statements of the main results regarding problem (P λ ) will be shown in Section 2. For the sake of a clear presentation, we consider for now a model problem:

(1.1)

         -∆u + µ(x) |∇u| 2 (u + δ) γ = λu p , x ∈ Ω, u > 0, x ∈ Ω, u = 0, x ∈ ∂Ω,
where δ ≥ 0, γ > 0, p > 1 and µ ∈ L ∞ (Ω). Our main goal is to allow µ to be nonconstant and even sign-changing, paying special attention to the singular case δ = 0.

Problem (1.1) for µ ≡ 0 becomes semilinear; as p > 1, it is usually referred to as superlinear. This superlinear case is classical and has been extensively studied in the literature. Indeed, both variational (in [START_REF]IWTHA IPVS!IPVVF PV yF vdyzhenskyD xF rl9tsevD Linear and quasilinear elliptic equations. rnslted from the ussin y ript ehniD[END_REF]) and topological (in [21, 25]) methods can be used to prove the existence of a solution to (1.1) for all λ > 0 provided µ ≡ 0 and p ∈ (1, 2 * -1), where 2 * = 2N N -2 . It is well-known that Pohozaev's identity (see [36]) implies that the restriction p < 2 * -1 is necessary for the existence of solution to the superlinear problem if the domain is starshaped.

The study of problem (1.1) for a nontrivial µ was initiated in [35]. There, the authors consider p ∈ (1, 2 * -1), δ > 0 and µ ≡ constant > 0. In this setting, they prove several results which can be divided into two classes: on the one hand, those which lead to a set of solutions similar to that for the classical semilinear problem (i.e., existence for all λ > 0) and, on the other hand, those which present dierences such as nonexistence of solution for λ > 0 small. In fact, the semilinearlike behavior is achieved provided γ > 1, while the dierences appear if either γ ∈ (0, 1) or γ = 1 and µ > p. In several proofs, the authors of [35] make use of the change of variable

(1.2) v = ψ(u) = u 0 e -s 0 µ (t+δ) γ dt ds.
It is easy to check formally that, if µ ≡ constant and δ > 0, then the transformation (1.2) turns (1.1) into a semilinear problem in the variable v. Thus, roughly speaking, the gradient term is removed and semilinear techniques (such as variational methods) can be applied in general. However, such a transformation can be performed only if µ is constant.

We remark that problem (1.1) in the case γ = 1 is specially interesting because of the condition µ > p that appears in [35], which shows that the interaction between the gradient term and the superlinear term in (1.1) plays an important role. To this respect, some results concerning the case γ = 1 (always with δ > 0 and µ ≡ constant) that improve those in [35] in some directions can be found in [3,29]. In the rst work, the authors prove nonexistence of solution for every λ > 0 small enough provided µ ≥ p, while in the second one the authors prove existence of solution for every λ > 0 provided µ < 2 * -1-p 2 * -2 . We point out that, in both mentioned works, no restriction on p from above is imposed. Nevertheless, if p ≥ 2 * -1, then the condition µ < 2 * -1-p 2 * -2 required by the existence result in [29] forces µ to be negative. We also stress that a blow-up argument is employed in [29] in order to obtain a priori estimates, even though the change of unknown (1.2) is strongly used in order to get rid of a quadratic gradient term from the general problem that the authors consider and, in consequence, non-constant functions µ cannot be handled with their approach.

Still focusing on problem (1.1) with γ = 1, δ > 0 and µ ≡ constant, the range 2 * -1-p 2 * -2 ≤ µ < p has not been considered in the literature to our knowledge. However, in this particular situation it is easy to see that the transformation (1.2) turns (1.1) into a semilinear equation whose nonlinearity presents supercritical growth at innity and subcritical growth at zero. Therefore, [7, Theorem 8] implies (after undoing the change of unknown) that there exists at least a solution to (1.1) for every λ > 0 large enough. Again, last (immediate) result is based on the change of unknown, so does not cover problem (1.1) with nonconstant µ.

On the other hand, the singular case δ = 0 has been dealt with recently in [18], one more time for µ ≡ constant > 0 (see also [14,19] for similar singular problems that involve a nonzero source term). The authors of [18] show that, if γ ∈ (0, 1), then the situation is similar to the nonsigular case δ > 0. Indeed, they prove a nonexistence result for λ > 0 small and an existence result for λ > 0 large. On the contrary, for γ ≥ 1 they prove nonexistence results for all λ > 0. This fact exposes the remarkable inuence of a strong singularity in the equation. As far as we know, the µ ≡ constant case for δ = 0 has not been studied in the literature.

To sum up, in the present work we aim to develop an approach that permits to deal with x-dependent µ in problem (1.1) and also with singular lower order terms, i.e., δ = 0. In order to do so, we will employ topological methods. More precisely, we will nd solutions to (1.1) as xed points of certain compact operator that will be dened in Section 4. The well-denition of such an operator will require the well-posedness of the following problem:

(1.3)        -∆u + µ(x) |∇u| 2 u = h(x), x ∈ Ω, u > 0, x ∈ Ω, u = 0, x ∈ ∂Ω,
where 0 h ∈ L q (Ω) for some q > N 2 . Singular problems of this kind have risen interest in the recent years. In fact, the existence of solution with µ L ∞ (Ω) < 1 2 has been proven in [11], and extended to µ L ∞ (Ω) < 1 in [32]. As far as the uniqueness of solution is concerned, some results are known for problems similar to (1.3), even though they require either the singularity to be milder or µ to be constant (see [4,8,16]). In next result we show that uniqueness for problem (1.3) holds provided µ L ∞ (Ω) < 1; the proof is based on a comparison principle that we state in Section 2 and prove in Section 4. Furthermore, we will show that the condition µ L ∞ (Ω) < 1 is natural by proving a nonexistence result provided µ > 1 in a neighborhood of ∂Ω; the proof follows the ideas in [18,Lemma 2.5]. The statement of the result reads as follows.

Theorem 1.1. Let 0 h ∈ L q (Ω) for some q > N 2 and let 0 µ ∈ L ∞ (Ω). The following statements hold true:

(1) If µ L ∞ (Ω) < 1, then there exists a unique nite energy solution to problem (1.3).

(2) If there exist an open domain ω ⊂⊂ Ω and a constant τ > 1 such that µ(x) ≥ τ and h(x) = 0, both for a.e. x ∈ Ω \ ω, then problem (1.3) admits no solution.

We point out that existence and uniqueness results for problem (1.3) are known for general nonnegative µ ∈ L ∞ (Ω) (i.e., without assuming that µ L ∞ (Ω) < 1) provided h is locally bounded away from zero (see [2] for the existence and [16] for the uniqueness). Thus, we clarify that the condition h ≡ 0 near the boundary in Theorem 1.1 is also natural for having nonexistence of solution.

Once we have shown that problem (1.3) is well-posed, we will be able to dene a compact operator whose xed points are solutions to (P λ ) (see Section 4). A version of a result in [27] (see [21]) will assure the existence of a xed point of the operator.

As it is mandatory for xed point theorems, we will prove the existence of a priori estimates on the solutions to a problem related to (P λ ). To this task, we will adapt the blow-up method due to [25]. Roughly speaking, this technique consists of assuming by contradiction that there exists a sequence of solutions whose norms blow up as n tends to innity. The conclusion follows by passing to the limit in a problem satised by a certain normalized sequence. In fact, the limit function is a solution to a problem which, however, does not admit any solution by virtue of some Liouville type result. Therefore, one gets a contradiction, so any sequence of solutions must be bounded. In this context, the diculties that we nd are twofold. Firstly, the normalized sequence, say {v n }, satises an equation having a lower order term of type |∇vn| 2 vn+δn , where 0 ≤ δ n → 0 as n → +∞. If we aim to pass to the limit, then we need to nd positive lower bounds on {v n }, otherwise the lower order term may blow up as n → +∞. And lastly, we arrive to a limit problem, having a quadratic gradient lower order term of the form |∇v| 2 v , for which nonexistence Liouville type results are not known in the literature (a nonexhaustive list of references for Liouville type results about problems depending on the gradient is [23,24,30,34,3739]).

We overcome the rst of the diculties by proving Hölder estimates in spite of the singular quadratic term. The proof follows the ideas of [28], which have been widely used for singular problems (see [2,6,15,17,22,31,32], among others). We will show that these estimates yield in turn positive lower estimates from below and this will be enough to pass to the limit. Regarding the second diculty, we observe that the limit equation does admit a convenient change of unknown which gets rid of the gradient term, so that we may apply classical Liouville type results (see Section 3 below).

We state here the main existence result for problem (1.1) in the case γ = 1.

Theorem 1.2. Let p > 1, γ = 1, δ ≥ 0 and µ ∈ L ∞ (Ω).
The following statements hold true:

(1) If µ ∈ C(Ω) and there exist two real numbers τ, σ such that 2σ -

1 < τ ≤ σ < 2 * -1-p 2 * -2
and τ ≤ µ(x) ≤ σ for all x ∈ Ω, then there exists at least a solution to (1.1) for every λ > 0. If, in addition, µ ≥ 0, then there exists at least a nite energy solution to (1.1) for every λ > 0.

(2) If δ = 0 and there exist an open domain ω ⊂⊂ Ω and a constant τ > 1 such that µ(x) ≥ τ for a.e. x ∈ Ω \ ω, then problem (1.1) admits no solution for any λ > 0.

Note that, in the rst item of Theorem 1.2, µ is allowed to change sign unless p ≥ 2 * -1, in which case µ is necessarily negative. We also point out that the smallness condition σ < 2 * -1-p 2 * -2 in Theorem 1.2 is natural since, in fact, problem (1.1) has no bounded solutions provided γ = 1,

δ = 0, µ ≡ constant ∈ 2 * -1-p 2 * -2 , 1
and Ω is starshaped (see Remark 2.5 below). Moreover, we will show later that, strengthening the smallness condition conveniently (in terms of p, N ), one may assume µ to be either continuous only in a neighborhood of ∂Ω, or merely bounded in Ω. Also about the existence part of the theorem, it is worth to point out that we need to control µ from below in order to prove the Hölder estimates that we mentioned. However, if µ is constant, i.e., σ = τ , then the condition 2σ -1 < τ becomes σ = τ < 1, which means no restriction since σ < 2 * -1-p 2 * -2 < 1. On the other hand, concerning the nonexistence statement for the case δ = 0, we stress that the case µ ≡ 1 remains unsolved, i.e., there are neither existence nor non-existence results about problem (1.1)

for γ = 1, δ = 0, µ ≡ 1, p > 1 in the literature.
Next result shows that our approach allows also to prove existence for γ > 1 and for all λ > 0.

Theorem 1.3. Let p > 1, γ > 1, δ ≥ 0 and µ ∈ L ∞ (Ω).
The following statements hold true:

(1) If δ > 0 and µ satises the condition

(1.4) 2 µ + L ∞ (Ω) -µ - L ∞ (Ω) < δ γ-1 , µ + L ∞ (Ω) < δ γ-1 ,
then there exists at least a solution to (1.1) for every λ > 0. If, in addition, µ ≥ 0, then there exists at least a nite energy solution to (1.1) for every λ > 0.

(2) If δ = 0 and there exist an open domain ω ⊂⊂ Ω and a constant τ > 0 such that µ(x) ≥ τ for a.e. x ∈ Ω \ ω, then problem (1.1) admits no solution for any λ > 0.

Notice that, if µ ≡ constant > 0, then condition (1.4) becomes µ < δ γ-1 2 . It is known that such a smallness condition is not needed if µ is a positive constant (see [35,Theorem 1.2]). Therefore, we presume that the assumption (1.4) is technical, even though we cannot avoid it since it is used to prove the Hölder estimates mentioned above. However, (1.4) means no restriction if µ ≤ 0 in Ω. Concerning the nonexistence statement for the singular case δ = 0, it is proven again following closely [18,Lemma 2.5]. We stress that the fact that the singularity is strong, namely γ > 1, allows to take µ > 0 near ∂Ω (in contrast to the case γ = 1, for which µ > 1 near the boundary was needed).

We will be able to go beyond Theorem 1.2 and Theorem 1.3 and deal with any γ > 0. Indeed, for γ ∈ (0, 1) and δ ≥ 0, we will show that existence of solution holds for every λ > 0 large enough. We will also deal with γ ≥ 1, δ > 0 and µ a bounded function with arbitrary size, i.e., we remove the size restrictions on µ in Theorem 1.2 and Theorem 1.3 as long as δ > 0, even though λ must be large. The statement of the result is the following. Theorem 1.4

. Let p > 1, γ > 0, δ ≥ 0 and µ ∈ L ∞ (Ω). If γ ≥ 1, assume in addition that δ > 0.
Assume also one of the following two conditions:

(1

) p < N +1 N -1 , (2) p < 2 * -1 and 0 ≤ µ ∈ C(Ω).
Then, there exists λ 0 > 0 such that there exists at least a solution u λ to (1.1) for every λ > λ 0 satisfying lim λ→+∞ u λ L ∞ (Ω) = 0. If, in addition, µ ≥ 0, then u λ is a nite energy solution.

If γ ∈ (0, 1), last result is consistent with the known results for µ > 0 constant which assure nonexistence for λ > 0 small (see [35] for δ > 0 and [18] for δ = 0). On the contrary, as we pointed out above, for γ > 1 we would expect an existence result for every λ > 0 (even for λ small) without size restrictions on µ. As far as the case γ = 1 and δ > 0 is concerned, we ignore whether an existence result for λ > 0 small and general µ should be expected or not. Two exceptions are the ranges µ < 2 * -1-p 2 * -2 and µ ≥ p, for which existence (see Theorem 1.2 above and [29]) and nonexistence (see [4,35]) for λ > 0 small are known respectively. In other words, the existence of solution to (1.1) for γ = 1, δ > 0, 2 * -1-p 2 * -2 ≤ µ < p and λ > 0 small remains as an open problem, even for µ > 0 constant.

We organize the paper as follows. In Section 2 we introduce the conditions on f, g and the statements of the general results about problem (P λ ). Section 3 is devoted to proving some Liouville type results as well as several propositions that provide the estimates via the blow-up method. In Section 4 we prove the results that we state in Section 2 and in the Introduction. Finally, in the Appendix we gather some technical results that are required throughout the paper.

Acknowledgments. The problems considered in this work have been proposed by T. Leonori.

The research was initiated with his collaboration during one week in Granada and another week in Rome, when most of the ideas contained here emerged. Moreover, for the conclusion of the paper, the interesting ideas, comments and corrections by J. Carmona have also supposed a more than remarkable contribution. In any case, their supervision and support have been essential during the research period. This is why the author wants to warmly thank both, collaborators and friends, for being examples of altruism to follow in this competitive and sometimes hostile world of research. The author wants to thank also D. Ruiz and C. De Coster for their kind and useful suggestions and corrections.

Hypotheses and main results

Let us consider a continuous function f : [0, +∞) → [0, +∞) and a Carathéodory function g : Ω × (0, +∞) → R. We will assume throughout this paper that g satises the following condition:

(2.1) ∃g 0 ∈ C((0, +∞)) : |g(x, s)| ≤ g 0 (s) a.e. x ∈ Ω, ∀s > 0.

Last hypothesis is essentially the minimal condition that g must satisfy so that the weak formulation of (P λ ) is well dened:

Denition 2.1. Let f : [0, +∞) → [0, +∞) be a continuous function and g : Ω × (0, +∞) → R be a Carathéodory function satisfying (2.1). A solution to (

P λ ) is a function u ∈ H 1 loc (Ω)∩L ∞ (Ω) such that        ∀ω ⊂⊂ Ω, ∃c > 0 : u(x) ≥ c a.e. x ∈ ω, ∃γ > 0 : u γ ∈ H 1 0 (Ω),
and

Ω ∇u∇φ + Ω g(x, u)|∇u| 2 φ = Ω f (u)φ ∀φ ∈ C 1 c (Ω).
Besides, we will say that u is a nite energy solution to (P λ ) if it is a solution to (P λ ) with γ = 1,

i.e., if u ∈ H 1 0 (Ω).
Remark 2.2. It can be proven by following [15,Appendix] that, in the previous denition, one can take test functions φ ∈ H 1 0 (Ω) ∩ L ∞ (Ω) with compact support. Moreover, if u is a nite energy solution, then one can take any test function belonging to H 1 0 (Ω) ∩ L ∞ (Ω). Let us x σ ∈ R. The following condition, stronger than (2.1), means a more precise control on g from above:

(g 1 ) ∃g 0 ∈ C((0, +∞)) : -g 0 (s) ≤ g(x, s) ≤ σ s a.e. x ∈ Ω, ∀s > 0.

Let us consider also the monotonicity condition:

(g ) s → sg(x, s) is nondecreasing for a.e. x ∈ Ω.

We state next a comparison principle that will be the key for proving the uniqueness part of Theorem 1.1. The proof follows essentially the arguments in [15], which in turn are inspired by [5].

Theorem 2.3. Let 0 h ∈ L 1 loc (Ω) and let g : Ω × (0, +∞) → R be a Carathéodory function satisfying (g ) and (g 1 ) for some σ ∈ (0, 1).

Let u, v ∈ C(Ω) ∩ W 1,N loc (Ω), with u, v > 0 in Ω, be such that Ω ∇u∇φ + Ω g(x, u)|∇u| 2 φ ≤ Ω h(x)φ and (2.2) Ω ∇v∇φ + Ω g(x, v)|∇v| 2 φ ≥ Ω h(x)φ (2.3) for every 0 ≤ φ ∈ H 1 0 (Ω)∩L ∞ (Ω)
with compact support. Suppose also that the following boundary condition holds:

(2.4) lim sup x→x0 (u(x) 1-σ -v(x) 1-σ ) ≤ 0 ∀x 0 ∈ ∂Ω.
Then, u ≤ v in Ω.

We will present below four existence theorems that represent the core of this work. The proofs are based on a xed point result. The required a priori estimates are obtained via a blow-up method (see Section 3).

Let us x p > 1 and δ ≥ 0. We establish now a growth condition and two limit conditions on f which make it behave like a superlinear power:

(f * ) ∃a ≥ 1 : s p ≤ f (s) ≤ a(s + δ) p ∀s ≥ 0. (f ∞ ) ∃L ∈ (0, +∞) : lim s→+∞ f (s) s p = L. (f 0 ) lim s→0 f (s) s = 0.
Observe that (f 0 ) implies in particular that f (0) = 0.

On the other hand, for xed δ ≥ 0 and σ, τ ∈ R, we set the following growth restriction on g which will allow us to prove certain Hölder estimates (see the Appendix below):

(g * ) 2σ -1 < τ ≤ σ < 1, τ ≤ (s + δ)g(x, s) ≤ σ a.e. x ∈ Ω, ∀s > 0.
We will also need the following limit condition on g at innity for p > 1:

(g ∞ ) ∃µ ∈ C(Ω) : max x∈Ω µ(x) < 2 * -1-p 2 * -2 , lim s→+∞ sg(•, s) -µ L ∞ (Ω) = 0.
A simple limit condition on g at zero will be required too:

(g 0 ) ∃ lim s→0 (s + δ)g(x, s) a.e. x ∈ Ω.
Observe that, if δ > 0, then (g 0 ) is equivalent to g(x, •) being continuous at s = 0 for a.e. x ∈ Ω.

We are ready to state our rst general existence result:

Theorem 2.4. Let f : [0, +∞) → [0, +∞) be a continuous function and g : Ω × (0, +∞) → R be a Carathéodory function. For δ ≥ 0, σ, τ ∈ R and p > 1, assume that f satises (f * ), (f ∞ ) and (f 0 ), and that g satises (g * ),(g ∞ ) and (g 0 ). Then, there exists at least a solution to (P λ ) for every λ > 0. If, in addition, g ≥ 0, then there exists at least a nite energy solution to (P λ ) for every λ > 0.

Remark 2.5. We point out that the smallness condition max x∈Ω µ(x) < 2 * -1-p 2 * -2 in Theorem 2.4 coming from condition (g ∞ ) is necessary for the existence of solutions to (P λ ), at least in the particular case of Ω a satarshaped domain and sg(x, s) ≡ σ for some constant σ < 1. Indeed, assume by contradiction that σ ∈ 2 * -1-p 2 * -2 , 1 and that 0

< u ∈ H 1 0 (Ω) ∩ L ∞ (Ω) satises -∆u + σ |∇u| 2 u = λu p in Ω. Then, v = cu 1-σ satises -∆v = v p-σ 1-σ in Ω for some c > 0. Therefore, since p-σ 1-σ ≥ 2 * -1
and Ω is starshaped, the well-known Pohozaev's identity (see [36]) yields a contradiction.

Let us x again p > 1. Next hypothesis is a limit condition on g at innity weaker than (g ∞ ) as it is only required to hold in a neighborhood of ∂Ω.

( g ∞ ) ∃ω ⊂⊂ Ω, µ ∈ C(Ω \ ω) : max x∈Ω\ω µ(x) < 2 * -1-p 2 * -2 , lim s→+∞ sg(•, s) -µ L ∞ (Ω\ω) = 0.
The following result shows that, assuming a stronger control on g from above, one can relax the limit condition on g at innity to obtain a solution to (P λ ).

Theorem 2.6. Let f : [0, +∞) → [0, +∞) be a continuous function and g : Ω × (0, +∞) → R be a Carathéodory function. For δ ≥ 0, σ, τ ∈ R and p > 1, assume that f satises (f * ), (f ∞ ) and (f 0 ), and that g satises (g * ), ( g ∞ ) and (g 0 ). Assume in addition that σ ≤ N -(N -2)p 2 . Then, there exists at least a solution to (P λ ) for every λ > 0. If, in addition, g ≥ 0, then there exists at least a nite energy solution to (P λ ) for every λ > 0.

Next theorem shows that the limit conditions at innity are actually not essential to obtain solutions to (P λ ). In return, one has to assume an even stronger control on g from above.

Theorem 2.7. Let f : [0, +∞) → [0, +∞) be a continuous function and g : Ω × (0, +∞) → R be a Carathéodory function. For δ ≥ 0, σ, τ ∈ R and p > 1, assume that f satises (f * ) and (f 0 ), and that g satises (g * ) and (g 0 ). Assume in addition that σ ≤ N +1-(N -1)p 2 . Then, there exists at least a solution to (P λ ) for every λ > 0. Moreover, if δ = 0, then there exists C > 0 such that, for every λ > 0 and for every solution u to (P λ ), the estimate λ

1 p-1 u L ∞ (Ω) ≤ C holds. If, in addition, g ≥ 0,
then there exists at least a nite energy solution to (P λ ) for every λ > 0.

Let s 0 ∈ (0, 1), σ, τ ∈ R and p > 1. Consider the following growth conditions near zero:

( f * ) ∃a ≥ 1 : s p ≤ f (s) ≤ as p ∀s ∈ (0, s 0 ). ( g * ) 2σ -1 < τ ≤ σ < 1, τ ≤ sg(x, s) ≤ σ a.e. x ∈ Ω, ∀s ∈ (0, s 0 ).
Notice that condition ( f * ) implies that lim s→0 f (s) s = 0 and, in particular, f (0) = 0. On the other hand, we point out that, if τ > 0, then condition ( g * ) forces g to be singular at s = 0. However, if ( g * ) holds for some τ < 0 and σ > 0, then g may be rather general at s = 0 since it can be continuous as well as unbounded from above and from below.

Our next existence theorem will require neither growth nor limit conditions at innity on f and g. In exchange, the existence of solution holds only for every λ > 0 large enough.

Theorem 2.8. Let f : [0, +∞) → [0, +∞) be a continuous function and g : Ω × (0, +∞) → R be a Carathéodory function. For s 0 ∈ (0, 1), σ, τ ∈ R and p > 1, assume that f satises ( f * ) and that g satises ( g * ). Assume in addition that there exists lim s→0 sg(x, s) for a.e. x ∈ Ω, and also that σ ≤ N +1-(N -1)p 2 . Then, there exists λ 0 > 0 such that there exists at least a solution u λ to (P λ ) for every λ > λ 0 satisfying lim λ→+∞ u λ L ∞ (Ω) = 0. If, in addition, g ≥ 0, then u λ is a nite energy solution.

Regarding Theorem 2.8, observe that, if p > N +1

N -1 , then σ < 0, so g(x, s) < 0 for s near zero. Thus, for instance, the particular case g(x, s) = µ s γ and f (s) = s p is not covered by Theorem 2.8 if γ ∈ (0, 1), p ∈ N +1 N -1 , 2 * -1 and µ > 0 is constant. The result that comes next does cover this particular case. In fact, for p ∈ (1, 2 * -1), next theorem allows to consider general continuous functions satisfying

(G) ∃G ∈ L 1 ((0, s 0 )) : 0 ≤ g(x, s) ≤ G(s) ∀(x, s) ∈ Ω × (0, s 0 ). ( g 0 ) ∃µ 0 ∈ C(Ω) : max x∈Ω µ 0 (x) < 2 * -1-p 2 * -2 , lim s→0 sg(•, s) -µ 0 L ∞ (Ω) = 0.
Remark 2.9. Notice that (G) and ( g 0 ) imply that lim s→0 sg(•, s) L ∞ (Ω) = 0, i.e., µ 0 ≡ 0 in condition ( g 0 ). Indeed, we know that for every ε > 0 there exists s ε ∈ (0, 1) such that

sg(x, s) ≥ µ 0 (x) -ε ∀s ∈ (0, s ε ), ∀x ∈ Ω.
Let us assume by contradiction that there exists x 0 ∈ Ω such that µ 0 (x 0 ) > 0. Then we choose ε = µ0(x0) 2 and we obtain

G(s) ≥ g(x, s) ≥ ε s ∀s ∈ (0, s ε ).

This contradicts hypothesis (G).

The result reads as follows:

Theorem 2.10. Let g : Ω × (0, +∞) → R be a continuous function. For s 0 ∈ (0, 1) and p ∈

(1, 2 * -1), assume that g satises (G) and ( g 0 ). Let us also consider the function

f : [0, +∞) → [0 + ∞) dened by f (s) = s p for all s ≥ 0.
Then, there exists λ 0 > 0 such that there exists at least a nite energy solution u λ to (P λ ) for every λ > λ 0 satisfying lim λ→+∞ u λ L ∞ (Ω) = 0.

Remark 2.11. We point out that Theorem 2.10 is valid for a very wide class of nonlinearities.

For instance, if g(x, s) = µ(x)h(s) a.e. x ∈ Ω, ∀s ≥ 0, where 0 µ ∈ L ∞ (Ω) and h : [0, +∞) → [0, +∞) is continuous (also at s = 0), then g satises (G) and ( g 0 ). On the other hand, a prototypical example of function g singular at s = 0 satisfying the conditions (G) and ( g 0 ) is

g(x, s) = µ(x) s γ a.e. x ∈ Ω, ∀s > 0, where γ ∈ (0, 1), 0 µ ∈ L ∞ (Ω).
The last theorem of the section will be concerned with the following problem:

(H)      -∆u + g(x, u)|∇u| 2 = h(x), x ∈ Ω, u > 0, x ∈ Ω, u = 0, x ∈ ∂Ω.
The result provides a necessary condition for the existence of solution to (H). In order to prove it, we will assume, for xed τ > 0, that g(x, u) is suciently large near ∂Ω in the sense of the following condition:

(g 2 ) ∃ω ⊂⊂ Ω, s 0 ∈ (0, 1) : sg(x, s) ≥ τ a.e. x ∈ Ω \ ω, ∀s ∈ (0, s 0 ).
The theorem states the following:

Theorem 2.12. Let h ∈ L 1 (Ω) and let g : Ω × (0, +∞) → R be a Carathéodory function satisfying (2.1) and (g 2 ) for some τ > 1. Then, every solution u to (H) satises

Ω |h(x)| u = +∞.

A priori estimates

We begin the section with two Liouville type results which will be the key points for proving a priori estimates. The rst of them is the following. Lemma 3.1. Let p > 1 and let h : (0, +∞) → [0, +∞) be a continuous function satisfying sh(s) ≤ σ ∀s > 0 for some σ ∈ (0, 1). Let us assume that there exists α > 0 such that the function ψ : [0, +∞) → [0, +∞), dened by

ψ(s) = s 0 e -t α h(r)dr dt ∀s ≥ 0, satises that s → ψ (s)s p ψ(s) 2 * -1
is decreasing for all s > 0.

Then, the following problem

(3.1)      -∆u + h(u)|∇u| 2 = u p , x ∈ X, u > 0, x ∈ X, u = 0, x ∈ ∂X, admits no solutions in H 1 loc (X) ∩ C(X), where X denotes either R N or R N + .
Remark 3.2. We stress that Lemma 3.1 includes the particular case h(s) = σ s with σ < 2 * -1-p 2 * -2 . Indeed, in this case, it is easy to check that ψ (s)s p ψ(s 1) for some constant c > 0. Thus, it is decreasing if, and only if,

) 2 * -1 = cs p-σ-(1-σ)(2 * -
σ < 2 * -1-p 2 * -2 . Moreover, 2 * -1-p 2 * -2 < 1, so sh(s) = σ < 1 for all s > 0.
Proof of Lemma 3.1. Arguing by contradiction, assume that there exists a solution u ∈

H 1 loc (X)∩ C(X) to (3.1). Straightforward computations imply that v = ψ(u) ∈ H 1 loc (X) ∩ C(X) satises (3.2)      -∆v = ϕ(v), x ∈ X, v > 0, x ∈ X, v = 0, x ∈ ∂X, where ϕ(t) = ψ (ψ -1 (t))ψ -1 (t) p for all t ∈ Im(ψ). Moreover, classical elliptic regularity theory implies that v ∈ C 2 (X). We claim now that, actually, problem (3.2) admits no solutions v ∈ C 2 (X) ∩ C(X).
Indeed, one can easily deduce that

ψ(s) ≥ α σ s 1-σ -α 1-σ 1 -σ ∀s > α.
Hence, lim s→+∞ ψ(s) = +∞. In consequence, the function ϕ is dened in [0, +∞). On the other hand, it is clear that the function t → ϕ(t) t 2 * -1 is decreasing for all t > 0. Therefore, in case X = R N , the claim follows from [9, Theorem 3].

Notice also that lim t→0 ϕ(t) t = 0 and ϕ(t) ≥ cψ -1 (t) p-σ for every t > 0 and for some c > 0. Hence, the claim also holds true in case X = R N + by virtue of [20, Theorem 1.3]. In any case, we have arrived to a contradiction. The proof is now completed.

We present here the second Liouville type result which is valid for supersolutions.

Lemma 3.3. Let p > 1 and σ ≤ N -(N -2)p 2 . Then, the following problem

(3.3)        -∆u + σ |∇u| 2 u = u p , x ∈ X, u > 0, x ∈ X, u = 0, x ∈ ∂X, admits no supersolutions in H 1 loc (X) ∩ C(X) provided X = R N . On the other hand, if we assume that σ ≤ N +1-(N -1)p 2 , then problem (3.3) with X = R N + admits no supersolutions in H 1 loc (X) ∩ C(X). Remark 3.4. Note that N +1-(N -1)p 2 < N -(N -2)p 2 < 2 * -1-p 2 * -2 < 1
, so the smallness conditions on σ in Remark 3.2 and in Lemma 3.3 are gradually more restrictive. We also stress that such conditions on σ in Lemma 3.

3 are sharp. Indeed, if σ > N -(N -2)p 2 (resp. σ > N +1-(N -1)p 2 ), then one can nd explicit supersolutions to (3.3) for X = R N , see [33] (resp. X = R N + , see [10]).
Proof of Lemma 3.3. Reasoning by contradiction, assume that there exists

u ∈ H 1 loc (X) ∩ C(X) a supersolution to (3.3). Then, there is a constant c > 0 such that v = cu 1-σ ∈ H 1 loc (X) ∩ C(X) is a supersolution to      -∆v = v p-σ 1-σ , x ∈ X, v > 0, x ∈ X, v = 0, x ∈ ∂X. Hence, if σ ≤ N -(N -2)p 2 , then p-σ 1-σ ≤ N N -2 , so in case X = R N we arrived to a contraction with [33, Theorem 2.1]. On the other hand, if σ < N +1-(N -1)p 2 , then p-σ 1-σ < N +1 N -1
, so we have again a contradiction with [10, Theorem 3.1].

Let t ≥ 0, λ > 0, f : [0, +∞) → [0, +∞) be a continuous function satisfying (f * ) for some p > 1 and g : Ω × (0, +∞) → [0, +∞) be a Carathéodory function satisfying (g 1 ) for some σ ∈ (0, 1). From now and up to the end of the section, we will restrict ourselves to g ≥ 0 and we will also impose diverse upper bounds on p. However, we will show in Section 4 that, in most cases, those restriction can be relaxed for proving existence of solution.

Let us consider the following auxiliary problem:

(P t )      -∆u + g(x, u)|∇u| 2 = λf (u) + tu σ , x ∈ Ω, u > 0, x ∈ Ω, u = 0, x ∈ ∂Ω.
Note that, in the term tu σ , the exponent σ is the same number that appears in condition (g 1 ).

We will derive a priori estimates on the solutions to (P t ) which will provide the existence of solution to (P λ ).

Next proposition gives an a priori estimate on the parameter t in problem (P t ).

Proposition 3.5. Let λ > 0, f : [0, +∞) → [0, +∞) be a continuous function and g : Ω × (0, +∞) → [0, +∞) be a Carathéodory function. For p > 1 and σ ∈ (0, 1), assume that f satises (f * ) and g satises (g 1 ). Then, there exists t 0 > 0 such that problem (P t ) admits no solution for any t > t 0 .

Proof. Let u be a solution to (P t ) for some t > 0. For a xed smooth open set ω ⊂⊂ Ω, let λ 1 be the principal eigenvalue to the homogeneous Dirichlet eigenvalue problem in ω, and let ϕ 1 be any positive associated eigenfunction, i.e., λ 1 and ϕ 1 satisfy

     -∆ϕ 1 = λ 1 ϕ 1 , x ∈ ω, ϕ 1 > 0, x ∈ ω ϕ 1 = 0, x ∈ ∂ω. If we extend ϕ 1 ≡ 0 in Ω \ ω, then the function φ = ϕ1 u σ belongs to H 1 0 (Ω) ∩ L ∞ (Ω)
and has compact support. Taking φ as test function in (P t ) (which is allowed by virtue of Remark 2.2) we obtain

(3.4) Ω ∇u∇ϕ 1 u σ -σ Ω ϕ 1 |∇u| 2 u σ+1 + Ω g(x, u) ϕ 1 |∇u| 2 u σ = λ Ω f (u) u σ ϕ 1 + t Ω ϕ 1 .
On the one hand, it is clear by (g 1 ) that

-σ Ω ϕ 1 |∇u| 2 u σ+1 + Ω g(x, u) ϕ 1 |∇u| 2 u σ ≤ 0.
Thus, using also (f * ) we deduce from (3.4) that

(3.5) t Ω ϕ 1 + λ Ω u p-σ ϕ 1 ≤ Ω ∇u∇ϕ 1 u σ .
On the other hand, let us denote as ν the exterior normal unit vector to ∂ω. Then, Hopf's lemma implies that ν∇ϕ 1 < 0 on ∂ω. Hence, integration by parts in ω (-∆ϕ 1 )u 1-σ and Young's inequality yield

Ω ∇u∇ϕ 1 u σ = λ 1 ω ϕ 1 u 1-σ 1 -σ + ∂ω u 1-σ 1 -σ ν∇ϕ 1 < λ 1 ω ϕ 1 u 1-σ 1 -σ ≤ λ 2 Ω u p-σ ϕ 1 + C.
In sum, from (3.5) we deduce that t ≤ t 0 for some t 0 > 0, as we wanted to prove.

Next lemma provides a summability property that the solutions to (P t ) satisfy. The interesting point is that such a property becomes better as the singularity of g becomes stronger.

Lemma 3.6. Let λ > 0, σ ∈ (0, 1), f : [0, +∞) → [0, +∞) be a continuous function and g : Ω × (0, +∞) → [0, +∞) be a Carathéodory function satisfying (2.1). For some a > 0, p ≥ 1 and τ ∈ [0, 1), assume that

f (s) ≤ as p ∀s ≥ 0, sg(x, s) ≥ τ a.e. x ∈ Ω, ∀s > 0. Let u ∈ H 1 loc (Ω) ∩ L ∞ (Ω) be a solution to (P t ) for some t ≥ 0. Then, u γ ∈ H 1 0 (Ω) for every γ > max{1-σ,1-τ } 2 .
Proof. First of all, recall that Lemma 5.2 in the Appendix below implies that u ∈ C(Ω). Now, for any α > 0, let us consider the function

v = u α ∈ H 1 loc (Ω) ∩ C(Ω). It is easy to see that v satises -∆v = (1 -α -v 1 α g(x, v 1 α ))|∇v| 2 αv + αλv α-1 α f (v 1 α ) + αtv α-1+σ α ≤ (1 -α -τ )|∇v| 2 αv + αλav α-1+p α + αtv α-1+σ α , x ∈ Ω.
Moreover, if we take α ≥ max{1 -σ, 1 -τ }, then

(3.6) -∆v ≤ C, x ∈ Ω,
for some constant C > 0.

For every ε, β > 0, let us now consider the function φ

= (v β -ε) + . It is clear that φ ∈ H 1 loc (Ω) ∩ C(Ω).
Furthermore, the continuity up to ∂Ω implies that φ has compact support in Ω. In sum, φ ∈ H 1 0 (Ω) ∩ C(Ω) and has compact support. Therefore, even though v might not belong to H 1 0 (Ω), it follows from Remark 2.2 that one can take φ as test function in (3.6). Thus, we obtain

Ω χ {v β ≥ε} |∇v| 2 v 1-β ≤ C
, for another constant C > 0 independent of ε. Now we let ε tend to zero and, by virtue of Fatou's lemma, we deduce

Ω |∇v| 2 v 1-β ≤ C. Taking into account that |∇v| 2 v 1-β = C ∇v 1+β 2 2 = C ∇u α 1+β 2 2
, then we have proved that u α 1+β 2 ∈ H 1 0 (Ω) for every α ≥ max{1 -σ, 1 -τ } and every β > 0. This proves the result.

In the following result we prove a priori estimates on the solutions to (P t ) if f satises (f * ) and (f ∞ ) and g satises (g * ) and (g ∞ ). In the proof we adapt the blow-up method due to [25].

Proposition 3.7. Let λ > 0, f : [0, +∞) → [0, +∞) be a continuous function and g : Ω × (0, +∞) → [0, +∞) be a Carathédory function. For δ, τ ≥ 0, σ > 0 and p ∈ (1, 2 * -1), assume that f satises (f * ) and (f ∞ ), and that g satises (g * ) and (g ∞ ). Then, there exists C > 0 such that u L ∞ (Ω) ≤ C for every solution u to (P t ) for all t ∈ [0, t 0 ], where t 0 > 0 is given by Proposition 3.5.

Proof. Reasoning by contradiction, we assume that there exist two sequences {t n } ⊂ [0, t 0 ] and {u n } such that u n is a solution to (P tn ) for all n and u n L ∞ (Ω) → +∞ as n → +∞. By virtue of Lemma 5.4, we may consider a sequence {x n } ⊂ Ω satisfying

u n L ∞ (Ω) = u n (x n ) ∀n, x n → x 0 ∈ Ω, up to a subsequence.
We divide the rest of the proof into two parts. In the rst of them we consider the case x 0 ∈ Ω, while the second one is devoted to the case x 0 ∈ ∂Ω. In turn, we divide each part into several steps. CASE 1) x 0 ∈ Ω.

Step 1.1) Scaling.

Denoting d = dist(x 0 , ∂Ω)/2 > 0 and η n = u n -p-1 2 L ∞ (Ω) , we dene v n : B d/ηn (0) → [0, +∞) by v n (y) = η 2 p-1 n u n (x n + η n y) ∀y ∈ B d/ηn (0). Therefore, v n ∈ H 1 (B d/ηn (0)) ∩ L ∞ (B d/ηn (0)
) and satises the equation

(3.7) -∆v n + u n g n (y, u n ) |∇v n | 2 v n = λv p n f (u n ) u p n + t n η 2(p-σ) p-1 n v σ n , y ∈ B d/ηn (0),
where g n (y, s) = g(x n + η n y, s) for a.e. y ∈ B d/ηn (0) and s > 0.

Moreover, v n L ∞ (B d/ηn (0)) = v n (0) = 1.
Our aim now is to pass to the limit in (3.7). In the next step we will prove the a priori estimates that will provide such a limit.

Step 1.2) A priori estimates.

Let us x R > 0 and denote ω = B R (0). It is clear that ω ⊂ B d/ηn (0) for every n suciently large, so v n satises the equation (3.7) in ω and v n L ∞ (ω) = 1 for n large. Of course, the same thing happens in B 2R (0). Notice also that, if δ = 0, then Lemma 3.6 implies that u

γ n ∈ H 1 0 (Ω) for all γ ∈ max{1-σ,1-τ } 2 , 1 -σ = 1-τ
2 , 1 -σ , where this last equality follows from τ < σ in condition (g * ). Therefore, v γ n ∈ H 1 (B 2R (0)) for all γ ∈ 1-τ 2 , 1 -σ . This fact, together with conditions (g * ) and (f * ), allow to apply Lemma 5.5 in the Appendix below to deduce that there exist C > 0, α ∈ (0, 1) such that

v n C 0,α (ω) ≤ C
for every n large enough. As a consequence, there exists v ∈ C(ω) such that, up to a subsequence,

v n → v uniformly in ω.
Observe that v L ∞ (ω) = 1 so, in particular, v ≡ 0.

On the other hand, let us consider a function φ ∈ C 1 c (B 2R (0)) such that 0 ≤ φ ≤ 1 in B 2R (0) and φ ≡ 1 in ω. Now we multiply both sides of (3.7) by v n φ 2 ∈ H 1 0 (B 2R (0)) ∩ L ∞ (B 2R (0)) and integrate by parts, obtaining

B 2R (0) |∇v n | 2 φ 2 + 2 B 2R (0) v n φ∇v n ∇φ ≤ C,
where we have used that g ≥ 0, v n L ∞ (B 2R (0)) = 1 and, one more time, condition (f * ). Hence, by Young's inequality we easily deduce that

ω |∇v n | 2 ≤ C B 2R (0) |∇φ| 2 v 2 n + 1 ≤ C.
That is to say, v n H 1 (ω) ≤ C, and then, up to a subsequence,

v n v weakly in H 1 (ω).
We will prove next that, for all ω 0 ⊂⊂ ω, v n is bounded from below in ω 0 by a positive constant independent of n. The approach by comparison due to [32] is valid here. Indeed, it is straightforward to see that the function

w n = v 1-σ n 1-σ ∈ H 1 (ω) ∩ L ∞ (ω) satises ω ∇w n ∇φ = λ ω v p-σ n f (u n ) u p n φ + ω (σ -u n g n (y, u n )) |∇v n | 2 φ v σ+1 n + t n η 2(p-σ) p-1
n ω φ for all φ ∈ C 1 c (ω). Therefore, using conditions (g * ) and (f * ), we derive

ω ∇w n ∇φ ≥ λ ω v p-σ n φ for all 0 ≤ φ ∈ C 1 c (ω).
On the other hand, let z n ∈ H 1 0 (ω) ∩ C(ω) be the unique solution to

-∆z n = λv p-σ n , y ∈ ω, z n = 0, y ∈ ∂ω.
It is clear that z n → z uniformly in ω and z n z weakly in

H 1 0 (ω) for some z ∈ H 1 0 (ω) ∩ C(ω). In consequence, z satises -∆z = λv p-σ , y ∈ ω, z = 0, y ∈ ∂ω.
Since v 0 in ω, the strong maximum principle implies that, for every ω 0 ⊂⊂ ω, there exists c > 0 such that z ≥ c in ω 0 . Besides, by comparison, w n ≥ z n in ω. Hence, the uniform convergence of {z n } implies that, for every ε > 0, we may take n large enough so that v

1-σ n ≥ (1 -σ)(z -ε) in ω. In sum, by choosing ε = c 2 we conclude that (3.8) ∀ω 0 ⊂⊂ ω, ∃c ω0 > 0 : v n ≥ c ω0 , y ∈ ω 0 , ∀n large.
From the previous estimates, it is straightforward to prove that {∆v n } is bounded in L 1 loc (ω). Then, [12] implies that, passing to a subsequence,

∇v n → ∇v, y ∈ ω.
We are ready now to pass to the limit in (3.7).

Step 1.3) Passing to the limit.

We already know that v n ≥ c ω0 > 0 in ω 0 ⊂⊂ ω. Moreover, u n L ∞ (ω) → +∞. Therefore, u n → +∞ locally uniformly in ω. Then, by (g ∞ ) we deduce that |u n g n (y, u n ) -µ(x n + η n y)| → 0 locally uniformly in ω. In consequence, the continuity of µ yields u n g n (y, u n ) → µ(x 0 ) locally uniformly in ω.

In sum, we have that

0 ≤ u n g n (y, u n ) |∇v n | 2 v n → µ(x 0 ) |∇v| 2 v pointwise in ω.
Furthermore, it follows from conditions (f ∞ ) and (f * ) and from the Dominated Convergence Theorem that

ω v p n f (u n ) u p n φ → ω v p φ ∀φ ∈ C 1 c (ω).
Let us take φ ∈ C 1 c (ω) such that φ ≥ 0 as test function in the weak formulation of (3.7). By virtue of Fatou's lemma and using the convergences that we have proved, it is immediate to show that

ω ∇v∇φ + µ(x 0 ) ω |∇v| 2 v φ ≤ λ ω v p φ.
If we take now vφ vn ∈ H 1 0 (ω) ∩ L ∞ (ω) as test function in (3.7), we obtain

ω ∇v n ∇v v n φ - ω v|∇v n | 2 v 2 n φ + ω v v n ∇v n ∇φ + ω u n g n (y, u n ) v|∇v n | 2 v 2 n φ = λ ω v p-1 n v f (u n ) u p n φ + t n η 2(p-σ) p-1 n ω vφ v 1-σ n .
Observe that

0 ≤ (1 -σ) v|∇v n | 2 φ v 2 n ≤ (1 -u n g n (y, u n )) v|∇v n | 2 φ v 2 n → (1 -µ(x 0 )) |∇v| 2 v φ.
Then, again by Fatou's lemma we derive

ω ∇v∇φ + µ(x 0 ) ω |∇v| 2 v φ ≥ λ ω v p φ. That is to say, v ∈ H 1 (ω) ∩ C(ω) satises -∆v + µ(x 0 ) |∇v| 2 v = λv p , y ∈ ω.
Step 1.4) Conclusion.

Since ω = B R (0) for arbitrary R > 0, then a standard diagonal argument (see [25]) implies that v is well-dened in R N , it belongs to

H 1 loc (R N ) ∩ C(R N ) and it satises -∆v + µ(x 0 ) |∇v| 2 v = λv p , y ∈ R N .
Furthermore, it is straightforward to check that the function w = λ

1 p-1 v satises -∆w + µ(x 0 ) |∇w| 2 w = w p , y ∈ R N .
This is impossible by virtue of Lemma 3.1 (see also Remark 3.2).

CASE 2) x 0 ∈ ∂Ω.

Step 2.1) Scaling.

Recall that we are assuming that there exist sequences {t n } ⊂ [0, t 0 ], {u n } and {x n } ⊂ Ω such that u n is a solution to (P tn ) for all n, u n L ∞ (Ω) = u n (x n ) → +∞ as n → +∞ and x n → x 0 for some x 0 ∈ ∂Ω. Taking advantage of the smoothness of ∂Ω, we are allowed to perform a convenient change of coordinates in such a way that u n is a solution to a similar problem except that ∂Ω becomes at near x 0 (see Lemma 5.1 in the Appendix below for the detailed proof). In other words, we may assume without loss of generality that u n is a solution to (R tn ) for all n, where

(R t )      -div(M (x)∇v) + b(x)∇v + g(x, v)M (x)∇v∇v = λf (v) + tv σ , x ∈ Ω, v > 0, x ∈ Ω, v = 0, x ∈ Γ, with Ω ⊂ R N + , ∅ = Γ ⊂ ∂Ω ∩ ∂R N + and it is connected, M ∈ C 1 (Ω) N ×N and it is uniformly elliptic, and b ∈ C(Ω) N .
It is clear that

d n = dist(x n , ∂Ω) = dist(x n , Γ) =
x n,N for all n large enough. Arguing as in the previous case, we dene

v n (y) = η 2 p-1 n u n (x n + η n y) ∀y ∈ Ω n ∪ Γ n , where η n = u n -p-1 2 L ∞ (Ω) , 0 ∈ Ω n = B d/ηn (0)∩{y N > -d n /η n } and Γ n = B d/ηn (0)∩{y N = -d n /η n } for some d > 0.
It is easy to see that v n is well-dened for all n large enough and it satises

               -div(M n (y)∇v n ) + b n (y)∇v n + g n (y, u n )u n M n (y)∇v n ∇v n v n = λv p n f (u n ) u p n + η 2(p-σ) p-1 n t n v σ n , y ∈ Ω n , v n > 0, x ∈ Ω n , v n = 0, y ∈ Γ n , where M n (y) = M (x n + η n y), b n (y) = η n b(x n + η n y) and g n (y, •) = g(x n + η n y, •). Now, if {d n /η n }
is unbounded, we can extract a subsequence such that d n /η n → +∞ as n → +∞. In this case, n∈N Ω n = R N , so we can argue as in the case x 0 ∈ Ω without relevant changes and arrive to a contradiction.

Let us assume now that {d n /η n } is bounded. Then, up to a (not relabeled) subsequence, d n /η n → κ for some κ ≥ 0. Thus, n∈N Ω n = {y N > κ}. Let us prove the estimates in this new situation.

Step 2.2) A priori estimates.

Let us denote

T n = (0, ..., 0, d n /η n ) ∈ R N , Ω n = B d/ηn (T n ) ∩ R N + , Γ n = B d/ηn (T n ) ∩ ∂R N + . Observe that n∈N Ω n = R N + and n∈N Γ n = ∂R N + . Let us dene w n : Ω n ∪ Γ n → [0, +∞) by w n (y) = v n (y -T n ) ∀y ∈ Ω n ∪ Γ n .
Obviously, w n satises (3.9)

                     -div(M n (y)∇w n ) + η n b n (y)∇w n + g n y, w n η 2 p-1 n 1 η 2 p-1 n M n (y)∇v n ∇v n = λη 2p p-1 n f w n η 2 p-1 n + η 2(p-σ) p-1 n t n w σ n , y ∈ Ω n , w n > 0, x ∈ Ω n , w n = 0, y ∈ Γ n , where M n (y) = M n (y -T n ), b n (y) = b n (y -T n ) and g n (y, •) = g n (y -T n , •). We also have that w n L ∞ (Ω n ) = w n (T n ) = 1 for all n.
Let us denote T ∞ = (0, ..., 0, κ) ∈ R N and, for R > κ, let us consider the sets

ω = B R (T ∞ ) ∩ R N + , β = B R (T ∞ ) ∩ ∂R N + ω 1 = B 2R (T ∞ ) ∩ R N + , β 1 = B 2R (T ∞ ) ∩ ∂R N + .
It is easy to see that, for all n large enough,

T n ∈ ω ⊂ ω 1 ⊂ Ω n , 0 ∈ β ⊂ β 1 ⊂ Γ n , ω ⊆ ω 1 ∪ β 1 .
In particular, w n satises problem (3.9) by changing Ω n with ω 1 and Γ n with β 1 , and further, w n L ∞ (ω1) = 1 for all n large enough. Thus, it follows again from Lemma 5.5 in the Appendix (thanks to conditions (g * ) and (f * ) and to Lemma 3.6) that there exist C > 0, α ∈ (0, 1) such that w n C 0,α (ω) ≤ C for every n large enough. As a consequence, there exists v ∈ C(ω) such that, up to a subsequence, w n → v uniformly in ω. In particular, v = 0 on β.

On the other hand, from the Hölder estimate one can also deduce that κ > 0. Indeed,

1 = |w n (T n ) -w n (0)| ≤ C|T n | α = C(d n /η n ) α → Cκ α .
Next, an estimate on {w n } in H 1 loc (ω) can be proven as in CASE 1, so that w n v weakly in H 1 loc (ω), up to a subsequence. Moreover, the same arguments are valid to prove that {w n } satises also (3.8) and, furthermore, that ∇w n → ∇v a.e. in ω.

Step 2.3) Passing to the limit.

We can now pass to the limit as in CASE 1 and deduce that v ∈ H 1 loc (ω) ∩ C(ω) is a solution to the following problem:

       -div(M (x 0 )∇v) + µ(x 0 ) M (x 0 )∇v∇v v = λv p , y ∈ ω, v > 0, y ∈ ω, v = 0, y ∈ β.
Actually, as R is arbitrary, the diagonal argument (see [25])

implies that v ∈ H 1 loc (R N + ) ∩ C R N + and it satises        -div(M (x 0 )∇v) + µ(x 0 ) M (x 0 )∇v∇v v = λv p , y ∈ R N + , v > 0, y ∈ R N + , v = 0, y ∈ ∂R N + .
Step 2.4) Conclusion.

Observe that, if we denote M (x 0 ) = (m ij ) for i, j = 1, ..., N , then the previous equation may be written as

N i,j=1 m ij ∂ 2 v ∂y i ∂y j + µ(x 0 ) v N i,j=1
m ij ∂v ∂y i ∂v ∂y j = λv p , y ∈ R N + .

Since i, j commute in both ∂ 2 v ∂yi∂yj and ∂v ∂yi ∂v ∂yj , then a simple change of coordinates (see the conclusion of Case 1 in Section 2 of [25] for the details) leads to nding a solution w ∈ H

1 loc (R N + )∩ C R N + to        -∆w + µ(y 0 ) |∇w| 2 w = w p , y ∈ R N + , w > 0, y ∈ R N + , w = 0, y ∈ ∂R N
+ , for some y 0 ∈ ∂Ω. This contradicts Lemma 3.1 (see also Remark 3.2). The proof is concluded.

The following result, as Proposition 3.7, provides a priori estimates on the solutions to (P t ). The dierence lies on the fact that we do not impose the limit conditions at innity (g ∞ ) and (f ∞ ) at the expense of making a stronger restriction on σ, p.

Proposition 3.8. Let λ > 0, f : [0, +∞) → [0, +∞) be a continuous function and g : Ω × (0, +∞) → [0, +∞) be a Carathéodory function. For δ, τ ≥ 0, σ > 0 and p ∈ 1, N +1 N -1 , assume that f satises (f * ) and g satises (g * ). Assume in addition that σ ≤ N +1-(N -1)p 2 . Then, there exists C > 0 such that u L ∞ (Ω) ≤ C for every solution u to (P t ) for all t ∈ [0, t 0 ], where t 0 > 0 is given by Proposition 3.5.

Proof. The proof is very similar to that of Proposition 3.7. Here we give only a sketch.

Arguing by contradiction, we assume that there exist two sequences {t n } ⊂ [0, t 0 ] and {u n } such that u n is a solution to (P tn ) for all n and u n L ∞ (Ω) → +∞ as n → +∞. We also consider a sequence {x n } ⊂ Ω satisfying

u n L ∞ (Ω) = u n (x n ) ∀n, x n → x 0 ∈ Ω, up to a subsequence;
this can be done by virtue of Lemma 5.4. We denote

η n = u n -p-1 2 L ∞ (Ω)
. Let us assume that x 0 ∈ ∂Ω (we omit the simpler case x 0 ∈ Ω). Arguing as in the proof of Proposition 3.7, we assume without loss of generality that u n is a solution to (R tn ) for all n, with Ω = V ⊂ R N + and Γ ∂Ω ∩ ∂R N + . Thus, there exists a sequence of bounded domains {Ω n } satisfying, for every n, that 0 ∈ Ω n , x n + η n y ∈ Ω for all y ∈ Ω n and n∈N Ω n = X, where X may be either R N or {y n > κ} for some κ ≥ 0.

In any case, we dene

v n : Ω n → R by v n (y) = η 2 p-1 n u n (x n + η n y) ∀y ∈ Ω n .
It is easy to prove that v n satises the equation

-div(M n (y)∇v n ) + b n (y)∇v n + u n g n (y, u n ) M n (y)∇v n ∇v n v n = λv p n f (u n ) u p n + η 2(p-σ) p-1 n t n v σ n , y ∈ Ω n ,
where M n (y) = M (x n + η n y), b n (y) = η n b(x n + η n y) and g n (y, •) = g(x n + η n y, •). Since sg n (x, s) ≤ σ, M n (y)∇v n ∇v n ≥ 0 and f (u n ) ≥ u p n , we deduce that

-div(M n (y)∇v n ) + b n (y)∇v n + σ M n (y)∇v n ∇v n v n ≥ λv p n , y ∈ Ω n .
Passing to the limit in the previous inequality as in Proposition 3.7, and applying after that a convenient change of coordinates, we nd a supersolution v ∈ H 1 loc (X) ∩ C(X) to (3.3), where either X = R N or X = R N + . This is a contradiction with Lemma 3.3.

Next proposition provides similar estimates as Proposition 3.7 and Proposition 3.8. The novelty is that g satises now a limit condition at innity only in a neighborhood of ∂Ω. This means that we need to impose stronger restrictions on σ, p than in Proposition 3.7, but milder than in Proposition 3.8. Proposition 3.9. Let λ > 0, f : [0, +∞) → [0, +∞) be a continuous function and g : Ω × (0, +∞) → [0, +∞) be a Carathéodory function. For δ, τ ≥ 0, σ > 0 and p ∈ 1, N N -2 , assume that f satises (f * ) and (f ∞ ), and that g satises (g * ) and ( g ∞ ). Assume in addition that σ ≤ N -(N -2)p 2 . Then, there exists C > 0 such that u L ∞ (Ω) ≤ C for every solution u to (P t ) for all t ∈ [0, t 0 ], where t 0 > 0 is given by Proposition 3.5.

Proof. We argue similarly as for Proposition 3.7 and Proposition 3.8, so we give a sketch of the proof.

Assume by contradiction that there exist two sequences {t n } ⊂ [0, t 0 ] and {u n } such that u n is a solution to (P tn ) for all n and u n L ∞ (Ω) → +∞ as n → +∞. Thanks to Lemma 5.4, we may also consider a sequence {x n } ⊂ Ω satisfying

u n L ∞ (Ω) = u n (x n ) ∀n, x n → x 0 ∈ Ω up to a subsequence.
Suppose that x 0 ∈ Ω. Since x 0 might belong to ω (where we know nothing about the asymptotic behavior of g at innity), we cannot proceed as in the proof of Proposition 3.7. Nevertheless, scaling u n conveniently and using (g * ) and (f * ) we may argue as in the proof of Proposition 3.8 to nd a supersolution 0

< v ∈ H 1 loc (R N ) ∩ C(R N ) to -∆v + σ |∇v| 2 v = v p , y ∈ R N .
This is a contradiction with Lemma 3.3.

On the other hand, if x 0 ∈ ∂Ω, then we may take advantage of ( g ∞ ) to obtain, using also (f ∞ ) and arguing as in Proposition 3.7, a solution v ∈ H 1 loc (X) ∩ C(X) to (3.3) for σ = µ(y 0 ) and some y 0 ∈ Ω, and either X = R N or X = R N + . This contradicts Lemma 3.1 (see also Remark 3.2).

Next result provides an estimate for the solutions to problem (P λ ) whose dependence on λ is explicit. As a consequence, it is shown that the norm of the solutions to problem (P λ ), if they exist, becomes arbitrarily small as λ tends to innity. Proposition 3.10. Let f : [0, +∞) → [0, +∞) be a continuous function and g : Ω × (0, +∞) → [0, +∞) be a Carathéodory function. For δ = 0, τ ≥ 0, σ > 0 and p ∈ 1, N +1 N -1 , assume that f satises (f * ) and g satises (g * ). Assume in addition that σ ≤ N +1-(N -1)p 2 . Then, there exists C > 0 such that

λ 1 p-1 u L ∞ (Ω) ≤ C
for every solution u to (P λ ) for all λ > 0.

Proof. Arguing again as in the proof of Proposition 3.7, assume that there exist two sequences {λ n } ⊂ [0, +∞) and {u n } such that u n is a solution to (P λn ) for all n and z n L ∞ (Ω) → +∞ as n → +∞, where z n = λ

1 p-1 n u n . It is easy to see that z n satises -∆z n + λ -1 p-1 n g x, λ -1 p-1 n z n |∇z n | 2 = λ p p-1 n f λ -1 p-1 n z n , x ∈ Ω.
Since z n ∈ C(Ω) by virtue of Lemma 5.4, then we may take {x n } ⊂ Ω such that z n (x n ) = z n L ∞ (Ω) for all n, Let x 0 ∈ Ω be such that, passing to a subsequence if necessary, x n → x 0 ∈ ∂Ω (the case x 0 ∈ Ω is analogous, so we omit it). Let us consider the function v n (y) = η 2 p-1 n z n (x n + η n y) dened for all y ∈ Ω n , with Ω n ⊂ R N as in the proof of Proposition 3.7 (Case 2) and

η n = z n -p-1 2 L ∞ (Ω)
. Then, thanks to Lemma 5.1 in the Appendix below, we may assume without loss of generality that

v n ∈ H 1 (Ω n ) ∩ L ∞ (Ω n ) satises (3.10) -div(M n (y)∇v n )+b n (y)∇v n + v n L n g n y, v n L n M n (y)∇v n ∇v n v n = L p n f v n L n , y ∈ Ω n ,
where L p-1 n = λ n η 2 n and M n , b n , g n are as in the proof of Proposition 3.7 (Case 2). From (3.10), and using conditions (f * ) and (g * ), one can prove the same estimates on {v n } as in the proof of Proposition 3.7. Moreover, again by (f * ) and (g * ) we deduce that

-div(M n (y)∇v n ) + b n (y)∇v n + σ M n (y)∇v n ∇v n v n ≥ v p n , y ∈ Ω n .
Now we pass to the limit as usual and obtain a supersolution v ∈ H 1 loc (X) ∩ C(X) to (3.3), where either X = R N or X = R N + . This is a contradiction by virtue of Lemma 3.3.

In the last result of this section we impose stronger restrictions on f and g to obtain an a priori estimate for the solutions to (P λ ) similar to that in Proposition 3.10. The advantage is that p can be chosen near 2 * -1.

Proposition 3.11. Let g : Ω × (0, +∞) → [0, +∞) be a continuous function. For δ = 0, τ ≥ 0, σ > 0 and p ∈ (1, 2 * -1), assume that g satises (g * ), (g ∞ ) and ( g 0 ). Assume in addition that there exists α > 0 such that, for each x 0 ∈ Ω and L > 0, the function ψ : [0, +∞) → [0, +∞), dened by

ψ(s) = s 0 e -1 L t αL g(x0, r L )dr dt ∀s ≥ 0, satises that (3.11) s → ψ (s)s p ψ(s) 2 * -1
is decreasing for all s > 0.

Lastly, let us consider the function f : [0, +∞) → [0, +∞) dened by f (s) = s p for all s ≥ 0.

Then, there exists C > 0 such that

λ 1 p-1 u L ∞ (Ω) ≤ C
for every solution u to (P λ ) for all λ > 0.

Proof. We may reproduce the proof of Proposition 3.10 up to (3.10), where {v n } satises the same estimates as in the proof of Proposition 3.7. Now, unlike in the proof of Proposition 3.10, we aim to pass to the limit directly in (3.10). In order to do so, let us take a not relabeled subsequence so that L n → L for some L ∈ [0, +∞]. Let us assume rst that L = 0. In this case, we can pass to the limit in (3.10) as in the proof of Proposition 3.7 and we obtain a solution v ∈ H 1 loc (X) ∩ C(X) to (3.3) with σ = µ(y 0 ) for some y 0 ∈ Ω, where either X = R N or X = R N + . This is a contradiction by virtue of Lemma 3.1 (see also Remark 3.2).

Therefore, necessarily L > 0. Assume now that L = +∞. Using ( g 0 ), we may pass to the limit again in (3.10) and we nd a solution v ∈ H 1 loc (X) ∩ C(X) to (3.3) with σ = µ 0 (y 0 ) for some y 0 ∈ Ω, where either X = R N or X = R N + . One more time, this contradicts Lemma 3.1 (see also Remark 3.2).

The only remaining possibility is 0 < L < +∞. Thanks to the continuity of g, we may pass to the limit in (3.10) once more to obtain a solution v ∈ H 1 loc (X) ∩ C(X) to (3.1), where h(s) = 1 L g y 0 , s L for some y 0 ∈ Ω and either X = R N or X = R N + . Again, this is a contradiction with Lemma 3.1.

Proofs of the main results

We start by proving Theorem 2.3.

Proof of Theorem 2.3. Let us rst dene the function

g : Ω × (0, +∞) → [0, +∞) by g(x, s) = σ -s 1 1-σ g x, s 1 1-σ (1 -σ)s
, a.e. x ∈ Ω, ∀s > 0.

Thanks to (g 1 ) and (g ), it is clear that g is nonincreasing in the s variable. Moreover,

|g(s, x)| ≤ σ + s 1 1-σ g 0 s 1 1-σ (1 -σ)s , a.e. x ∈ Ω, ∀s > 0,
so it is a bounded function in the x variable.

For some 0 ≤ φ ∈ H 1 0 (Ω) ∩ L ∞ (Ω) with compact support, let us take (1-σ)φ

u σ
as test function in (2.2). Then, if we denote ũ = u 1-σ , we deduce that

Ω ∇ũ∇φ ≤ Ω g(x, ũ)|∇ũ| 2 φ + Ω (1 -σ)h(x) ũ σ 1-σ φ.
Arguing similarly, ṽ = v 1-σ satises

Ω ∇ṽ∇φ ≥ Ω g(x, ṽ)|∇ṽ| 2 φ + Ω (1 -σ)h(x) ṽ σ 1-σ φ.
Moreover, ũ, ṽ ∈ C(Ω) ∩ W 1,N loc (Ω) and they satisfy lim sup x→x0 (ũ(x) -ṽ(x)) ≤ 0 for all x 0 ∈ ∂Ω. At this point one can reproduce the proof of [31, Theorem 2.1] without relevant changes and conclude that ũ ≤ ṽ in Ω. Equivalently, u ≤ v in Ω.

Next we prove Theorem 2.12.

Proof of Theorem 2.12. Let v be a solution to (H). Recall that the denition of solution implies that there exists γ > 0 such that v γ ∈ H 1 0 (Ω). It is easy to see that u = v γ satises the equation (4.1)

-∆u + g γ (x, u)|∇u| 2 = γv γ-1 h(x), x ∈ Ω,
where

g γ (x, s) = s 1 γ g x, s 1 γ + γ -1 γs , a.e. x ∈ Ω, ∀s > 0. It follows from (g 2 ) that (4.2) sg γ (x, s) ≥ τ + γ -1 γ a.e. x ∈ Ω \ ω, ∀s ∈ (0, s γ 0 ),
where clearly τ +γ-1 γ > 1. From now we will denote ρ = τ +γ-1 γ and s 1 = s γ 0 . Recall that there exists c > 0 such that v ≥ c 1 γ in ω. For every ε ∈ (0, min{s 1 , c}), let us dene the function

ϕ ε : [0, +∞) → [0, +∞) by ϕ ε (s) =      1 s + s ρ-1 -ε ρ-1 (ρ -1)s ρ , s ≥ ε, s ε 2 , 0 ≤ s < ε. Clearly, ϕ ε (u) ∈ H 1 0 (Ω) ∩ L ∞ (Ω).
Hence, taking ϕ ε (u) as test function in the weak formulation of (4.1) (this can be done thanks to Remark 2.2 because u ∈ H 1 0 (Ω)) we obtain

(4.3) Ω ϕ ε (u)|∇u| 2 + Ω g γ (x, u)|∇u| 2 ϕ ε (u) = γ Ω v γ-1 h(x)ϕ ε (u).
Proof of Theorem 2.4. We divide the proof into several steps.

Step 1) An auxiliary problem

In addition to the hypotheses of Theorem 2.4, let us further assume for now that p < 2 * -1 and g 0. At the end of the proof of the theorem we will make the extension to general p, g.

Let us take q > N p 2 . For every 0 v ∈ L q (Ω), t ≥ 0, λ > 0, we consider the following problem:

(Q t )        -∆u + (v + δ)g(x, v) |∇u| 2 u + δ = λf (v) + tv σ , x ∈ Ω, u > 0, x ∈ Ω, u = 0, x ∈ ∂Ω,
where σ is the same real number that appears in condition (g * ); note that σ ∈ (0, 1) since g 0.

It is clear that, if u is a solution to (Q t ) with t = 0 and v = u, then it is actually a solution to (P λ ). In this step we will show that, if v 0, there exists a unique nite energy solution to (Q t ).

Observe that, if 0 v ∈ L q (Ω), then 0 λf (v) + tv σ ∈ L q/p (Ω), where q p > N 2 . Therefore, if δ = 0, then Theorem 1.1 implies directly that there exists a unique nite energy solution to (Q t ), that we will denote as u 0 . On the other hand, let us assume that δ > 0. Then, u 0 is clearly a subsolution to (Q t ). Let us consider the unique solution w ∈ H

1 0 (Ω) ∩ C(Ω) to the linear problem      -∆w = λf (v) + tv σ , x ∈ Ω, w > 0, x ∈ Ω, w = 0, x ∈ ∂Ω.
Then, w is a supersolution to (Q t ). Furthermore, u 0 ≤ w in Ω by comparison, so [13, Théorème 3.1] implies that there exists a nite energy solution

u δ to (Q t ) such that 0 < u 0 ≤ u δ ≤ w in Ω.
Moreover, every solution to (Q t ) belongs to C(Ω) ∩ W 1,N loc (Ω) by virtue of Lemma 5.4, Lemma 5.2 and Remark 5.3. Also, the function (x, s) → (v(x)+δ)g(x,v(x)) s+δ satises (g 1 ) and (g ). Therefore, Theorem 2.3 implies that u δ is the unique solution to (Q t ).

Step 2) Existence of a nite energy solution to (P λ ) if p < 2 * -1 and g 0.

Let X = {w ∈ C(Ω) : w(x) ≥ 0 ∀x ∈ Ω}. We have shown in the previous step that the operator K : X × [0, +∞) → X given by K(v, t) = u, ∀v ∈ X \ {0}, t ≥ 0, where u is the unique solution to (Q t ), K(0, t) = 0, ∀t ≥ 0. is well-dened. We aim to prove that there exists 0 u ∈ X such that K(u, 0) = u, which is equivalent to nding a solution to (P λ ).

We rst prove that K is continuous. Indeed, let {v n } ⊂ X and {t n } ⊂ [0, +∞) be such that

v n → v in C(Ω) for some v ∈ X and t n → t for some t ≥ 0. Let us denote u n = K(v n , t n ) and h n = λf (v n ) + t n v σ n . Taking into account that g ≥ 0, it is clear that (4.6) -∆u n ≤ h n , x ∈ Ω, ∀n.
We know that there exists C > 0 such that h n ≤ C for all n. Then, (4.6) leads to -∆u n ≤ C in Ω.

From this inequality, it is straightforward to prove that {u n } is bounded in H 1 0 (Ω). Hence, up to a subsequence, u n u weakly in H 1 0 (Ω) and u n → u strongly in L m (Ω) for some 0 ≤ u ∈ H 1 0 (Ω) and for all m ∈ [1, 2 * ). Furthermore, the inequality (4.6) yields the following estimate by virtue of the well-known Stampacchia's lemma (see [40]):

(4.7) u n L ∞ (Ω) ≤ C h n L ∞ (Ω) , ∀n.
In particular, if v ≡ 0, (4.7) yields directly that u n → 0 strongly in C(Ω). That is to say, K is continuous in {0} × [0, +∞).

Let us assume on the contrary that v 0. In this case, we will prove following [32] that {u n } is locally bounded away from zero. Indeed, rst of all observe that, using that h n ≤ C, we deduce from (4.7) that {u n } is bounded in L ∞ (Ω). Hence, using (f * ), (g * ) and the L ∞ estimate on u n , one can easily check that the function

w n = u 1-σ n satises -∆w n ≥ (1 -σ)λv p n u σ n ≥ Cv p n , x ∈ Ω,
for some constant C > 0. On the other hand, let z n ∈ H 1 0 (Ω) ∩ C(Ω) be the unique solution to the linear problem

-∆z n = Cv p n , x ∈ Ω, z n = 0, x ∈ ∂Ω.
By comparison, w n ≥ z n for all n. Moreover, it is clear that {z n } is bounded in H 1 0 (Ω) and, following [32] (which in turn is based on [28]), we also have that {z n } is bounded in C 0,α (Ω) for some α ∈ (0, 1). Hence, passing to a subsequence, z n converges weakly in H 1 0 (Ω) and strongly in

C(Ω) to the unique solution z ∈ H 1 0 (Ω) ∩ C(Ω) to -∆z = Cv p , x ∈ Ω, z = 0, x ∈ ∂Ω.
Since v 0, the strong maximum principle implies that, for all ω ⊂⊂ Ω, there exists c ω > 0 such that z ≥ 2c ω in ω. Furthermore, since z n → z uniformly in Ω, it follows that z n ≥ z -c ω in Ω for all n large enough. In sum,

u 1-σ n = w n ≥ z n ≥ c ω , x ∈ ω,
as we claimed.

It is clear now that {∆u n } is bounded in L 1 loc (Ω). Hence, from [12] we deduce that ∇u n → ∇u a.e. in Ω. We may now pass to the limit using Fatou's lemma twice, similarly as in the proof of Proposition 3.7 (Case 1, Step 1.3), so that u is the unique solution to (Q t ) i.e., u = K(v, t).

It remains to prove that u n → u strongly in C(Ω). Indeed, since {u n } is locally bounded away from zero, Lemma 5.2 implies that, for every ω ⊂⊂ Ω, one may take a subsequence such that u n → u strongly in C(ω). Actually, the uniqueness of u assures that the original sequence {u n }, and not merely a subsequence, converges itself to u strongly in C(ω). On the other hand, let

ζ n ∈ H 1 0 (Ω) ∩ C(Ω) be the unique solution to -∆ζ n = h n , x ∈ Ω, ζ n = 0, x ∈ ∂Ω. By comparison, u n ≤ ζ n in Ω for all n. Moreover, ζ n → ζ strongly in C(Ω), where ζ ∈ H 1 0 (Ω) ∩ C(Ω) is the unique solution to -∆ζ = λf (v) + tv σ , x ∈ Ω, ζ = 0, x ∈ ∂Ω.
In particular, for xed ε > 0, we deduce that

|u n (x) -u(x)| ≤ ζ n (x) + u(x) < ζ(x) + u(x) + ε 2 ∀x ∈ Ω, ∀n large enough.
Therefore, since u, ζ ∈ C(Ω) with u = ζ = 0 on ∂Ω, there exits ω ⊂⊂ Ω such that

|u n (x) -u(x)| < ε ∀x ∈ Ω \ ω, ∀n large enough.
Since u n → u strongly in C(ω), we conclude that

|u n (x) -u(x)| < ε ∀x ∈ Ω, ∀n large enough.
In other words, u n → u strongly in C(Ω), so K is continuous.

We prove now that K is compact, i.e., it maps bounded sets to relatively compact sets. Indeed, let {v n } ⊂ X and {t n } ⊂ [0, +∞) be bounded sequences. Taking subsequences, v n → v in the weak-topology of L ∞ (Ω) for some v ∈ L ∞ (Ω), while t n → t for some t ≥ 0. This is enough to pass to the limit in the equations as above. We conclude that, up to subsequences, K(v n , t n ) → K(v, t) strongly in C(Ω). This proves that K is compact.

We will prove next that there exist 0 < r < R and t 1 ≥ 0 such that

(1) u = sK(u, 0) ∀s ∈ [0, 1], ∀u ∈ X with u L ∞ (Ω) = r, (2) u = K(u, t) ∀t ≥ 0, ∀u ∈ X with u L ∞ (Ω) = R, (3) u = K(u, t) ∀t ≥ t 1 , ∀u ∈ X with u L ∞ (Ω) ≤ R.
In order to prove item ( 1), let us assume by contradiction that, for all r > 0, there exist s ∈ [0, 1] and u ∈ X with u L ∞ (Ω) = r such that u = sK(u, 0). In particular, u > 0 in Ω and

-∆u ≤ sλf (u), x ∈ Ω, u = 0, x ∈ ∂Ω.
Since (f 0 ) holds, we can choose r > 0 such that λf (t) ≤ λ1 2 t for all t ∈ [0, r], where λ 1 stands for the principal eigenvalue of -∆ in Ω with zero Dirichlet boundary conditions. Furthermore, u ≤ r in Ω, so we have that

       -∆u ≤ λ 1 2 u, x ∈ Ω, u > 0, x ∈ Ω, u = 0, x ∈ ∂Ω.
This is a clear contradiction with the denition of λ 1 .

On the other hand, if we take R > C, where u L ∞ (Ω) ≤ C (see Proposition 3.7) then it is clear that (2) holds. Moreover, if we take t 1 > t 0 , where t 0 > 0 is given by Proposition 3.5, then (3) also holds.

In conclusion, [21, Proposition 2.1 and Remark 2.1] can be applied and in consequence we obtain a positive xed point of K 0 , i.e., a nite energy solution to (P λ ).

Step 3) Extension to the general case From this point we assume only the hypotheses of Theorem 2.4, i.e., g need not be non-negative and p might be greater than or equal to 2 * -1.

Fix λ > 0. For some γ > 1 that will be chosen later, let us consider the following problem:

(4.8)      -∆v + g γ (x, v)|∇v| 2 = λf γ (s), x ∈ Ω, v > 0, x ∈ Ω, v = 0, x ∈ ∂Ω,
where

f γ (s) = b(s + δ γ ) γ-1 γ f (s + δ γ ) 1 γ -δ , ∀s ≥ 0, g γ (x, s) = (s + δ γ ) 1 γ g x, (s + δ γ ) 1 γ -δ + γ -1 γ(s + δ γ ) , a.e. x ∈ Ω, ∀s > 0,
and b > 0 is a constant, which depends only on p, δ, γ, that will be chosen below too. It is easy to see that, if v is a solution to (4.8), then u = v 1 γ is a solution to (P λ ). We will now choose γ > 1 so that problem (4.8) admits a nite energy solution.

Indeed, let us rst denote p γ = γ-1+p γ . Obviously, p γ > 1. It is easy to see that the function

s → s pγ (s + δ γ ) γ-1 γ (s + δ γ ) 1 γ -δ p is bounded in [0, +∞).
Then, using that f satises (f * ), b > 0 can be chosen (depending only on p, δ, γ) such that f γ (s) ≥ s pγ ∀s ≥ 0. Moreover, since f satises (f * ), it is clear that f γ (s) ≤ abγ(s + δ γ ) pγ ∀s ≥ 0.

We have proved that f γ satises (f * ). Taking into account that f satises (f ∞ ) and (f 0 ), it is straightforward to check that also f γ satises (f ∞ ) and (f 0 ) respectively.

On the other hand, since g satises (g ∞ ), then

lim s→+∞ sg γ (•, s) - µ + γ -1 γ L ∞ (Ω) = 0.
Using that max x∈Ω µ(x) < 2 * -1-p 2 * -2 , it is easy to see that

max x∈Ω µ(x) + γ -1 γ < 2 * -1 -p γ 2 * -2 .
Thus, g γ satises (g ∞ ). It is also immediate to check that g γ satises (g 0 ) using that so does g. Notice nally that, since g satises (g * ), then

τ + γ -1 γ ≤ (s + δ γ ) g γ (s, x) ≤ σ + γ -1 γ a.e. x ∈ Ω, ∀s > 0.
It is clear that

2 σ + γ -1 γ -1 < τ + γ -1 γ ≤ σ + γ -1 γ < 1.
Therefore, g γ satises (g * ).

If we now choose γ > 1 large enough so that τ +γ-1 γ > 0 and p γ < 2 * -1, then we may apply the previous step in order to obtain a nite energy solution to (4.8). The proof is now nished.

Proofs of Theorem 2.6 and Theorem 2.7. The proofs are analogous to that for Theorem 2.4. The only essential dierence lies in how the estimates on the solutions are obtained. In fact, instead of Proposition 3.7, we have to apply Proposition 3.9 for Theorem 2.6 and both Proposition 3.8 and Proposition 3.10 for Theorem 2.7.

Here we carry out the proof of Theorem 2.8.

Proof of Theorem 2.8. Let us dene the Carathéodory function ḡ : Ω × (0, +∞) → R by

ḡ(x, s) =    g(x, s) if x ∈ Ω, s ∈ (0, s 0 ), s 0 g(x, s 0 ) s if x ∈ Ω, s ≥ s 0 .
Let us also consider the continuous function f : [0, +∞) → [0, +∞) dened by

f (s) =    f (s) if s ∈ (0, s 0 ), f (s 0 ) s p 0 s p if s ≥ s 0 .
It is clear that f and ḡ satisfy the hypotheses of Theorem 2.7. Thus, if we write ( Pλ ) for denoting problem (P λ ) with ḡ and f instead of g and f , then Theorem 2.7 implies that there exists a solution u λ to ( Pλ ) for all λ > 0 that satises

λ 1 p-1 u λ L ∞ (Ω) ≤ C ∀λ > 0.
In particular, there exists λ 0 > 0 such that u λ L ∞ (Ω) < s 0 for every λ ≥ λ 0 . Hence, u λ is, in fact, a solution to (P λ ) for every λ ≥ λ 0 . The proof is nished.

It is left to prove Theorem 2.10.

Proof of Theorem 2.10. Let σ ∈ 0, min

1 2 , 2 * -1-p 2 * -2
. By virtue of Remark 2.9, we may take

δ ∈ (0, 1) such that sg(x, s) ≤ σ ∀(x, s) ∈ Ω × (0, δ].
Notice that δ can be chosen as small as necessary. In fact, it will be taken small enough several times throughout the proof, depending its size only on N, p, G.

Let us dene the function ḡ : Ω × (0, +∞) → [0, +∞) by

ḡ(x, s) =    g(x, s) if (x, s) ∈ Ω × (0, δ], δg(x, δ) s if (x, s) ∈ Ω × (δ, +∞).
It is clear that ḡ satises (g * ), (g ∞ ) and (g 0 ). Thus, if we write ( Pλ ) for denoting problem (P λ ) with ḡ instead of g, then Theorem 2.4 implies that there exists a solution to ( Pλ ) for all λ > 0. We aim now to prove that the hypotheses of Proposition 3.11 hold true, so that there exists C > 0 such that (4.9)

λ 1 p-1 u L ∞ (Ω) ≤ C ∀u solution to ( Pλ ) with λ > 0.
Observe that, if this is true, in particular there exists λ 0 > 0 such that u L ∞ (Ω) ≤ δ for every solution u to ( Pλ ) with λ ≥ λ 0 . Hence, the solutions to ( Pλ ) with λ ≥ λ 0 are, in fact, solution to (P λ ). Thus, the proof will nish as soon as we prove (4.9).

We will now prove that (3.11) in Proposition 3.11 holds. Indeed, for any L > 0, x 0 ∈ Ω, let us consider the function ψ : [0, +∞) → [0, +∞) given by ψ(s) = We will prove next that s → ψ (s)s p ψ(s) 2 * -1 is decreasing for all s > 0, or equivalently, that (4.10) (p -sḡ(x 0 , s)) ψ(sL) L < (2 * -1)sψ (sL) ∀s > 0.

Notice that neither ψ(sL) L nor ψ (sL) depends on L. In order to prove (4.10), we distinguish two possibilities: either 0 < s ≤ δ or s > δ. Assume rst that 0 < s ≤ δ. Then,

(p -sḡ(x 0 , s)) ψ(sL) L -(2 * -1)sψ (sL) =(p -sg(x 0 , s)) s 0 e -t δ g(x0,r)dr dt -(2 * -1)se -s δ g(x0,r)dr .
It follows from the mean value theorem that there exists θ ∈ (0, 1) such that Using that lim δ→0 δ 0 G(r)dr = 0 and p < 2 * -1, we are allowed to choose δ small enough so that pe δ 0 G(r)dr < 2 * -1.

Hence, (4.10) holds for 0 < s ≤ δ.

We assume now that s > δ. In this case,

ψ(s) = AL + δ τ L s L 1-τ -δ 1-τ 1 -τ ,
where τ = δg(x 0 , δ) and

A = ψ(δL) L = δ 0 e δ t g(x0,r)dr dt. Recall that τ ≤ σ < 2 * -1-p 2 * -2 . Then, p -τ 1 -τ ≤ p -σ 1 -σ < 2 * -1. Let us denote γ = 2 * -1 -p-σ 1-σ . We deduce that (p -sḡ(x 0 , s)) ψ(sL) L -(2 * -1)sψ (sL) = (p -τ ) A + δ τ s 1-τ -δ 1-τ 1 -τ -(2 * -1)δ τ s 1-τ ≤ (2 * -1) (1 -τ )A + δ τ (s 1-τ -δ 1-τ ) -δ τ s 1-τ -γ A + δ τ s 1-τ -δ 1-τ 1 -τ ≤ (2 * -1)(A -δ) -γA = A (2 * -1) 1 - δ A -γ ≤ A   (2 * -1)   1 -δ δ 0 e δ t G(r)dr dt -1   -γ   .
It is clear that

lim δ→0 δ 0 e δ t G(r)dr dt δ = lim δ→0 e δ 0 G(r)dr δ 0 e -t 0 G(r)dr dt δ = lim δ→0 e -δ 0 G(r)dr = 1.
Then, we can take δ small enough so that

(2 * -1)   1 -δ δ 0 e δ t h(r)dr dt -1   < γ.
In conclusion, (4.10) holds for s > δ, and thus, for all s > 0.

In sum, it follows that the hypotheses of Proposition 3.11 are fullled for problem ( Pλ ) so our claim was true. The proof is now nished.

We conclude the section by proving Theorem 1.2, Theorem 1.3 and Theorem 1.4 as consequences of the general results in Section 2.

Proof of Theorem 1.2. The rst item of the theorem is a direct consequence of Theorem 2.4, while the second one follows from Theorem 2.12.

Proof of Theorem 1.3. Let us denote g(x, s) = µ(x) (s+δ) γ . In order to prove the rst item, we assume that δ > 0 and that µ satises (1.4). It is easy to see that inf

(x,s)∈Ω×[0,+∞) (s + δ)g(x, s) = µ - L ∞ (Ω) δ γ-1 , sup (x,s)∈Ω×[0,+∞) (s + δ)g(x, s) = µ + L ∞ (Ω) δ γ-1 . Therefore, τ ≤ (s + δ)g(x, s) ≤ σ for τ = µ - L ∞ (Ω) δ γ-1 and σ = µ + L ∞ (Ω) δ γ-1
. Hence, (1.4) implies that g satises (g * ). On the other hand, it is clear that (s + δ)g(•, s) L ∞ (Ω) → 0 as s → +∞. Thus, g also satises (g ∞ ). In conclusion, Theorem 2.4 implies that there exists a solution to (1.1) for any λ > 0.

On the contrary, let us assume now that δ = 0 and µ(x) ≥ τ > 0 for a.e. x ∈ Ω \ ω. Then, (g 2 ) holds. Thus, Theorem 2.12 implies that problem (1.1) admits no solution for any λ > 0.

Proof of Theorem 1.4. Let us denote g(x, s) = µ(x) (s+δ) γ . We know that, for every ε > 0, there exists s 0 > 0 such that

(4.11) s|g(x, s)| ≤ µ L ∞ (Ω) s (s + δ) γ < ε ∀s ∈ (0, s 0 ), a.e. x ∈ Ω.
Therefore, ( g 0 ) holds (in particular, lim s→0 sg(x, s) obviously exists for a.e. x ∈ Ω). If we assume that 0 µ ∈ C(Ω) and p < 2 * -1, then the hypotheses of Theorem 2.10 are fullled, so we conclude.

Let us assume on the contrary that p < N +1 N -1 . Then, in (4.11) we may take σ = -τ = ε ∈ 0, min 1 3 , N +1-(N -1)p 2 in such a way that g satises ( g * ). Thus, the result follows after applying Theorem 2.8.

The proof is nished.

5. Appendix 5.1. Technical lemma. In this subsection we prove a technical result that is required by the blow-up method.

Lemma 5.1. Let Ω ⊂ R N (N ≥ 3) be a bounded domain with boundary of class C 2 , λ > 0, g : Ω × (0, +∞) → R be a Carathéodory function, and f : [0, +∞) → R be a continuous function.

Then, for every x 0 ∈ ∂Ω, there exist U ⊂ R N a neighborhood of x 0 and an injective and

C 2 map y : U → R N , with C 2 inverse, such that V = y(U ∩ Ω) ⊂ R N + , Γ = y(U ∩ ∂Ω) ⊂ ∂V ∩ ∂R N + and, if u is a solution to (P λ ), then the function v = u • y -1 : V → (0, +∞) is a solution to (5.1)      -div(M (y)∇v) + b(y)∇v + j(y, v)M (y)∇v∇v = f (v), y ∈ V, v > 0, y ∈ V, v = 0, y ∈ Γ, where M ∈ C 1 (V ) N ×N is uniformly elliptic, b ∈ C(V ) N and j(•, •) = g(y -1 (•), •), being M and b independent of u.
Proof. Since ∂Ω is of class C 2 , there exist U ⊂ R N a neighborhood of x 0 and a C 2 function

ψ : U → R, where U = {x ∈ R N -1 : ∃x N ∈ R, (x , x N ) ∈ U }, such that ψ(x ) < x N ∀(x , x N ) ∈ U ∩ Ω, ψ(x ) = x N ∀(x , x N ) ∈ U ∩ ∂Ω.
Let us dene the change of variables y : U → R N by

y(x) = (x , x N -ψ(x )) ∀x ∈ U.
It is clear that

y(x) ∈ R N + ∀(x , x N ) ∈ U ∩ Ω, y(x) ∈ ∂R N + ∀(x , x N ) ∈ U ∩ ∂Ω.
This proves that V ⊂ R N + and Γ ⊂ ∂R N + . Moreover, it is a simple exercise to prove that Γ ⊂ ∂V . It is also easy to see that the function y -1 : y(U ) → U given by y

-1 (z) = (z , z N + ψ(z )) ∀z ∈ y(U )
is the inverse function of y. Note that y -1 is well dened since z ∈ U for every z ∈ y(U ).

Let us now dene

v : V ∪ Γ → R by v(z) = u(y -1 (z)) ∀z ∈ V ∪ Γ.
Observe that v = 0 on Γ and u(x) = v(y(x)) for all x ∈ U ∩ Ω. We will show next that v satises an equation in V . Now we compute the derivatives that we need. We emphasize that such derivatives can be understood in a pointwise sense due to Remark 5.3 and to the C 2 regularity of ψ. We stress also that, as ψ does not depend on x N , it will be understood that ∂ψ

∂x N (x) = 0 for x ∈ U . a) Dy(x) =         1 0 • • • 0 0 0 1 . . . . . . . . . . . . 0 . . . 0 • • • 0 1 0 -∂ψ ∂x1 (x ) • • • • • • -∂ψ ∂x N -1 (x ) 1         , b)          ∂u ∂x i (x) = ∂v ∂z i (y(x)) - ∂v ∂z N (y(x)) ∂ψ ∂x i (x ), i = 1, • • • , N -1, ∂u ∂x N (x) = ∂v ∂z N (y(x)), c) |∇u(x)| 2 = |∇v(y(x))| 2 + ∂v ∂z N (y(x)) 2 |∇ψ(x )| 2 -2 ∂v ∂z N (y(x))∇v(y(x))∇ψ(x ), d) ∂ 2 u ∂x 2 i (x) = ∂ 2 v ∂z 1 ∂z i (y(x)) ∂y 1 ∂x i (x) + • • • + ∂ 2 v ∂z N ∂z i (y(x)) ∂y N ∂x i (x) - ∂v ∂z N (y(x)) ∂ 2 ψ ∂x 2 i (x ) - ∂ 2 v ∂z 1 ∂z N (y(x)) ∂y 1 ∂x i (x) + • • • + ∂ 2 v ∂z 2 N (y(x)) ∂y N ∂x i (x) ∂ψ ∂x i (x ) = ∂ 2 v ∂z 2 i (y(x)) -2 ∂ 2 v ∂z i ∂z N (y(x)) ∂ψ ∂x i (x ) + ∂ 2 v ∂z 2 N (y(x)) ∂ψ ∂x i (x ) 2 - ∂v ∂z N (y(x)) ∂ 2 ψ ∂x 2 i (x ), i = 1, • • • , N -1, e) ∆u = ∆v -2∇ ∂v ∂z N ∇ψ + ∂ 2 v ∂z 2 N |∇ψ| 2 - ∂v ∂z N ∆ψ.
Let us denote j(z, s) = g(y -1 (z), s) for a.e. z ∈ V and for all s > 0. Thus, v = v(y) (from this point y will simply denote variable in V ) satises the equation

-∆v - ∂ 2 v ∂y 2 N |∇ψ| 2 + 2∇ ∂v ∂y N ∇ψ + ∂v ∂y N ∆ψ = f (v) -j(y, v) |∇v| 2 + ∂v ∂y N 2 |∇ψ| 2 -2 ∂v ∂y N ∇v∇ψ , y ∈ V. (5.2)
Let us dene the matrix

M (y) =          1 0 • • • 0 -2 ∂ψ ∂x1 (y ) 0 1 . . . . . . . . . . . . 0 . . . 0 • • • 0 1 -2 ∂ψ ∂x N -1 (y ) 0 • • • • • • 0 1 + |∇ψ(y )| 2          , y ∈ V,
and also the vector b(y) = (0, • • • , 0, -∆ψ(y )), y ∈ V . Then, one can check that v is a solution to (5.1).

Moreover, let a > 1 be such that (a -1) ∇ψ 2 L ∞ (V ) < 1. Then, by Cauchy-Schwarz's and Young's inequalities, the following holds for every ξ ∈ R N :

M (y)ξξ = |ξ| 2 -2ξ N ∇ψξ + ξ 2 N |∇ψ| 2 ≥ |ξ| 2 -(a -1)ξ 2 N |∇ψ| 2 - 1 a |ξ | 2 = 1 - 1 a |ξ | 2 + (1 -(a -1)|∇ψ| 2 )ξ 2 N ≥ 1 - 1 a |ξ | 2 + (1 -(a -1) ∇ψ 2 L ∞ (V ) )ξ 2 N ≥ min 1 - 1 a , 1 -(a -1) ∇ψ 2 L ∞ (V ) |ξ| 2 .
Then, M is uniformly elliptic. The proof is nished.

Local Hölder estimate and global continuity.

This subsection is devoted to proving a local Hölder condition on the solutions to the problems that concern this manuscript. In particular, the solutions are continuous in the interior of Ω. We will show at the end of the section that the solutions are, in fact, continuous up to the boundary. Lemma 5.2. Let g : Ω × (0, +∞) → R be a Carathéodory function satisfying (2.1) and let h ∈ L q (Ω) for some q > N 2 . Then, for every M ≥ c > 0 and every ω ⊂⊂ Ω, there exist α ∈ (0, 1), L > 0 such that every solution u ∈ H 1 loc (Ω) ∩ L ∞ (Ω) to

(5.3) -∆u + g(x, u)|∇u| 2 = h(x), x ∈ Ω, c ≤ u ≤ M, x ∈ Ω,

satises

(5.4) u C 0,α (ω) ≤ L.

Proof. Let us consider m 0 = max s∈[c,M ] g 0 (s), where g 0 : (0, +∞) → [0, +∞) is the continuous function given by (2.1), and let us take δ ∈ 0, 1 2m0 . With the notation of [28, Chapter 2, Section 6, p. 81], we will show that u belongs to the class B 2 Ω, M, γ, δ, 1 2q for some γ > 0 to be determined later. This implies (5.4) by virtue of [28,Theorem 6.1,p. Clearly, (u -k) + ζ 2 ∈ H 1 0 (Ω) ∩ L ∞ (Ω) and has compact support in Ω, so it may be taken as test function in (5.3). Thus we obtain

2 A k,ρ ζ(u -k)∇ζ∇u + A k,ρ |∇u| 2 ζ 2 + A k,ρ g(x, u)|∇u| 2 (u -k)ζ 2 = A k,ρ h(x)(u -k)ζ 2 .
On the one hand, by (2.1) and by the denitions of k ρ , m 0 we deduce that

- A k,ρ g(x, u)|∇u| 2 (u -k)ζ 2 ≤ δm 0 A k,ρ |∇u| 2 ζ 2 .
On the other hand, Young's inequality yields

-2 A k,ρ ζ(u -k)∇ζ∇u ≤ 2 A k,ρ (u -k) 2 |∇ζ| 2 + 1 2 A k,ρ |∇u| 2 ζ 2 .
Therefore, we derive

1 2 -δm 0 A k,ρ |∇u| 2 ζ 2 ≤ 2 A k,ρ (u -k) 2 |∇ζ| 2 + A k,ρ h(x)(u -k)ζ 2 .
Next, it follows from Hölder's inequality that

A k,ρ h(x)(u -k)ζ 2 ≤ δ h L q (Ω) |A k,ρ | 1 q and 2 A k,ρ (u -k) 2 ζ 2 ≤ 2b ρ 2 σ 2 A k,ρ (u -k) 2 ≤ 2b ρ 2 σ 2 (u -k) 2 L ∞ (A k,ρ ) |A k,ρ | ≤ 2bC(N ) ρ 2-N q σ 2 (u -k) 2 L ∞ (A k,ρ ) |A k,ρ | 1 q ,
where |B ρ | = C(N ) q ρ N . In sum,

(5.5)

A k,ρ-σρ |∇u| 2 ≤ γ 1 ρ 2-N q σ 2 (u -k) 2 L ∞ (A k,ρ ) + 1 |A k,ρ | 1 q ,
where γ = max 2bC(N ), δ h L q (Ω) 1 2 -δm 0 .

Furthermore, notice that v = -u satises -∆v -g(x, -v)|∇v| 2 = -h(x), x ∈ Ω.

In sum, we may apply the arguments in [15,Appendix] without relevant changes to prove that u n C 0,α (ω) ≤ C for some C > 0, α ∈ (0, 1). In particular, using that the function s → s

  x0,r)dr dt ∀s ≥ 0.

  x0,r)dr dt -(2 * -1)se -s δ g(x0,r)dr =se δ s g(x0,r)dr (p -sg(x 0 , s))e s θs g(x0,r)dr -(2 * -1) . Thanks to (G) we derive (p -sg(x 0 , s))e s θs g(x0,r)dr -(2 * -1) ≤ pe δ 0 G(r)dr -(2 * -1).

  90]. Indeed, let us x ρ > 0 and let B ρ be an open ball of radius ρ such that B ρ ⊂ Ω. Let us x also σ ∈ (0, 1), and consider a function ζ ∈ C ∞ c (B ρ ) satisfying that 0 ≤ ζ ≤ 1 in B ρ , ζ ≡ 1 in the concentric open ball B ρ-σρ , and |∇ζ| < b σρ in B ρ for some constant b > 0 independent of ρ, σ. Let us denote k ρ = sup x∈Bρ u(x) -δ, and take k ≥ k ρ . We also consider the set A k,ρ = {x ∈ B ρ : u(x) ≥ k}.

1 γ

 1 is locally Lipschitz for s ≥ 0, we have that|v n (x) -v n (y)| = |v n (x) + δ n -v n (y) -δ n | ≤ C|(v n (x) + δ n ) γ -(v n (y) + δ n ) γ | = C|u n (x) -u n (y)| ≤ C|x -y| α ∀x, y ∈ ω.In conclusion, v n C 0,α (ω) ≤ C, as we wanted to prove.ReferencesI eF emrosettiD FrF inowitzD Dual variational methods in critical point theory and applications. J. Functional Analysis 14 @IWUQAD QRW!QVIF P hF eroyD tF grmonD F veonoriD FtF wrtínezEepriioD vF yrsinD pF ettitD Existence and nonexistence of solutions for singular quadratic quasilinear equationsF J. Dierential EquationsD 246 @PHHWAD RHHT!RHRPF Q hF eroyD tF grmonD FtF wrtínezEepriioD Bifurcation for quasilinear elliptic singular BVP. Comm. Partial Dierential Equations 36 @PHIIAD noF RD TUH!TWPF R hF eroyD tF grmonD F tF wrtínezEepriioD Comparison principle for elliptic equations in divergence with singular lower order terms having natural growth. Commun. Contemp. Math., 19 @PHIUAD ITSHHIQD II ppF S hF eroyD gF he gosterD vF tenjenD uF nkD Remarks on the uniqueness for quasilinear elliptic equations with quadratic growth conditions. J. Math. Anal. Appl. 420 @PHIRAD UUP!UVHF T hF eroyD vF worenoEwéridD The eect of a singular term in a quadratic quasi-linear problem. J. Fixed Point Theory Appl. 19 @PHIUAD VIS!VQIF U hF eroyD tFvF qámezD vF yrsinD sF erlD Local existence results for sub-super-critical elliptic problems. Commun. Appl. Anal. 5 @PHHIAD noF RD SSU!STWF V hF eroyD F egur de veónD Uniqueness of solutions for some elliptic equations with a quadratic gradient termF ESAIM Control Optim. Calc. Var., 16 @PHIHAD QPU!QQTF W qF finhiD Nonexistence of positive solutions to semilinear elliptic equations on R N or R N + through the method of moving planes. Comm. Partial Dierential Equations 22 @IWWUAD noF W!IHD ITUI!ITWHF IH sF firindelliD iF witidieriD Liouville theorems for elliptic inequalities and applications. Proc. Roy. Soc. Edinburgh Sect. A 128 @IWWVAD noF TD IPIU!IPRUF II vF fordoD Dirichlet problems with singular and gradient quadratic lower order terms. iesw gontrol yptimF glF rF 14 @PHHVAD RII!RPTF IP vF fordoD pF wurtD Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal. 19 @IWWPAD noF TD SVI!SWUF IQ vF fordoD pF wurtD tFEF uelD Quelques propriétés des opérateurs elliptiques quasi linéairesF C. R. Acad. Sci. Paris Sér. I Math. 307 @IWVVAD URW!USPF IR vF fordoD vF yrsinD wF orzioD Existence results for quasilinear elliptic and parabolic problems with quadratic gradient terms and sources. Adv. Calc. Var. 4 @PHIIAD noF RD QWUR!IWF IS tF grmonD F veonoriD F vópezEwrtínezD FtF wrtínezEepriioD Quasilinear elliptic problems with singular and homogeneous lower order termsF Nonlinear Anal. 179 @PHIWAD IHSEIQHF IT tF grmonD F veonoriD A uniqueness result for a singular elliptic equation with gradient termD Proc. Roy. Soc. Edinburgh Sect. A 148 @PHIVAD noF SD WVQ!WWRF IU tF grmonD F vópezEwrtínezD FtF wrtínezEepriioD Singular quasilinear elliptic problems with changing sign datum: existence and homogenization. Rev. Mat. Complut. 33 @PHPHAD noF ID QW!TPF IV tF grmonD FtF wrtínezEepriioD eF uárezD Existence and nonexistence of positive solutions for nonlinear elliptic singular equations with natural growthF Nonlinear Anal. 89 @PHIQAD ISU!ITWF IW tF grmonD eF wolinoD vF worenoEwéridD Existence of a continuum of solutions for a quasilinear elliptic singular problem. J. Math. Anal. Appl. 436 @PHITAD noF PD IHRV!IHTPF PH vF hmselliD pF qldiliD Some nonexistence results for positive solutions of elliptic equations in unbounded domains. Rev. Mat. Iberoamericana 20 @PHHRAD noF ID TU!VTF

Observe that, from (4.2), it follows that

On the other hand, it is clear that (4.5)

We will now absorb the negative term in (4.5) with the last term in (4.4). Indeed,

In conclusion, from (4.3), (4.5) and from the previous discussion we deduce that

)s for all s > 0, we nally deduce that

Therefore, we let ε tend to zero and by virtue of Fatou's lemma we obtain that

= +∞ and the proof is nished.

We are ready now to prove Theorem 1.1.

Proof of Theorem 1.1. The existence of a solution u ∈ H 1 0 (Ω) (not necessarily bounded) is a consequence of [32,Theorem 3.1]. Moreover, since q > N 2 , the well-known Stampacchia's lemma (see [40]) implies that u ∈ L ∞ (Ω).

On the other hand, Lemma 5.4 leads to u ∈ C(Ω). Furthermore, u ∈ W 1,N loc (Ω) by virtue of Lemma 5.2 and Remark 5.3 in the Appendix below. Therefore, the uniqueness of solution follows directly from Theorem 2.3.

Finally, it is clear that, if h 0 has compact support, then Ω h(x) u < +∞ for every solution u to (1.3). Thus, Theorem 2.12 implies the nonexistence part of the theorem. Now we prove Theorem 2.4. Since the proofs of Theorem 2.6 and Theorem 2.7 are analogous, we will only make some comments below.

Therefore, it is straightforward to check that the inequality (5.5) also holds by substituting u with -u. In conclusion, u ∈ B 2 Ω, M, γ, δ, 1 2q and the proof is nished.

Remark 5.3. Let g : Ω × (0, +∞) → R be a Carathéodory function satisfying (2.1), h ∈ L q (Ω)

Using the local Hölder regularity given by Lemma 5.2 one can prove that, if u ∈ H 1 loc (Ω) ∩ L ∞ (Ω) is a solution to (5.6) that is locally bounded away from zero, then u ∈ W 1,2q loc (Ω). The proof is based on a standard bootstrap argument, see [15,Appendix] for further details. Moreover, combining this local summability of the gradients with the classical Calderon-Zygmund regularity theory one can easily prove that, if h ∈ L ∞ (Ω), then u ∈ W 2,q loc (Ω) for every q < ∞. We will show next that every solution to (H) is continuous up to ∂Ω.

Lemma 5.4. Let g : Ω × (0, +∞) → [0, +∞) be a Carathédory function satisfying (2.1) and let h ∈ L q (Ω) for some q > N 2 . Then, every solution to (H) belongs to C(Ω). Proof. Let u ∈ H 1 loc (Ω)∩L ∞ (Ω) be a solution to (H). Clearly, Lemma 5.2 implies that u ∈ C(Ω). It is left to prove the continuity on the boundary. In order to do so, recall that there exists γ > 0 such that u γ ∈ H 1 0 (Ω). We may assume without loss of generality that γ > 1. Let us denote v = u γ . Using that g ≥ 0, we deduce that

for some constant C > 0. On the other hand, let us consider the unique solution w ∈ H 1 0 (Ω) ∩ C(Ω) to the following problem:

By standard comparison, 0 < v ≤ w in Ω. Therefore, lim x→y v(x) = 0 for every y ∈ ∂Ω and, in particular, u ∈ C(Ω).

Global Hölder estimates.

In the previous subsection we have shown that the solutions to the problems in this paper are locally Hölder continuous. In the proof, we have strongly used the fact that the solutions are locally bounded away from zero in order to avoid the possible singularity of g(x, u(x)) as x approaches ∂Ω. In the present subsection we aim to prove a global Hölder estimate, i.e., the singularity cannot be avoided since we need a Hölder estimate to hold even near ∂Ω. Therefore, we perform a dierent proof that requires g to satisfy the stronger condition (g * ).

Lemma 5.5. Let Ω ⊂ R N be a smooth bounded domain. Let {M n } ⊂ L ∞ (Ω) N ×N and {b n } ⊂ L ∞ (Ω) N be bounded sequences in the norm of their respective spaces. Assume that there exists η > 0 such that M n (y)ξξ ≥ η|ξ| 2 a.e. y ∈ ω, ∀ξ ∈ R N , ∀n. For some p > 1, δ, τ ≥ 0, σ ∈ (0, 1) independent of n, let f : [0, +∞) → [0, +∞) be a continuous function satisfying (f * ) and, for all n, let g n : ω × (0, +∞) → [0, +∞) be a Carathédory function satisfying (g * ). Let {v n } ⊂ H 1 (Ω) ∩ C(Ω) be such that 0 < v n ≤ C in Ω for all n and for some C > 0. For some sequences

If δ = 0, assume in addition that there exists γ ∈ 1-τ 2 , 1 -σ such that v γ n ∈ H 1 (Ω) for all n. Let us also assume that, for some connected (possibly empty) set Γ ⊂ ∂Ω,

Furthermore, if {L n } diverges, assume additionally that δ = 0 and n = 0 for all n. Then, for every smooth bounded domain ω satisfying that ω ⊆ Ω ∪ Γ, there exist α ∈ (0, 1), C > 0 such that, taking a (not relabeled) subsequence if necessary,

Proof. Let us denote δ n = L n δ and consider the function

2 , 1 -σ ; if δ = 0, then γ must be chosen so that u n ∈ H 1 (Ω) for all n. It is easy to see that u n ∈ C(Ω), 0 < u n ≤ C in Ω, u n = 0 on Γ, and it satises

where

Ln

g n y,

a.e. y ∈ Ω, ∀s > 0, ∀n,

a.e. y ∈ Ω, ∀n.

We claim that {u n fn (u n )} admits a subsequence bounded in L ∞ (Ω) and, furthermore,

(5.7) 0 ≤ sg n (y, s) ≤ c a.e. y ∈ Ω, ∀s > 0, ∀n, for some c ∈ (0, 1). In order to prove the claim, we rst observe that (f * ) implies

On the one hand, let us assume that {L n } is not divergent. Then, it admits a (not relabeled) bounded subsequence. Hence, taking into account that p > 1, it clearly holds that

Besides, since γ ≥ 1-σ 2 , then σ -1 + 2γ ≥ 0, so

Thus, {u n fn (u n )} is bounded in L ∞ (Ω). On the other hand, assume that {L n } diverges. Then, δ n = n = 0 for all n, so fn (u n ) ≤ au p-1+γ γ n ≤ C for all n and, again, {u n fn (u n )} is bounded in L ∞ (Ω).

Let us now verify that (5.7) holds. Indeed, from (g * ) and γ ∈ 1-τ 2 , 1 -σ , it follows

Thus, our claim holds.


Assessment of conformal use of personal protective equipment by object and human pose recognition

Le Bono Cdric 1,2 , Panagiotis Papadakis 1,2 , Buche Cdric 1,3

1 Lab-STICC, UMR 6285, F-29238, team RAMBO 2 IMT Atlantique Bretagne/Pays de la Loire, Brest, France 3 ENIB, Brest, France

Abstract-On construction sites, workers are exposed to a wide range of hazards. To reduce risk and prevent severe work accidents, the use of PPE (Personal Protective Equipment), such as hard hat or harness has been adopted by the industry. Monitoring the use of PPE is an inherently difficult task since its protective effectiveness depends on whether it is conformally worn. In recent years, the advent of deep learning for the tasks of object detection and human pose analysis has enabled the rise of discriminative and all-purpose PPE detectors. However, most of earlier works solely focus on detecting the equipment irrespective of their positioning. By assessing conformal PPE usage based on the combined output of object detection and articulated human pose extraction we can combine learning with domain expertise, which allows a straightforward, in-field application of the approach that attains high quantitative performance.

I. INTRODUCTION

According to the Occupational Safety and Health Administration (OSHA), in the USA, the construction industry suffered from 1008 worker fatalities, which represent 21.1% of the fatalities in private industries. The two main causes of death were falls and being struck by object, causing 338 and 112 deaths respectively [START_REF] Osha | Osha data and statistics[END_REF]. While controlling and trying to contain hazards at their origin is the best way to protect employees, this can only be done to an extent.

In this context and to alleviate the risk of accidents, regulations and safety legislation have been adopted, notably, regarding the use of personal protective equipment (PPE). Such equipment comes in various shapes and forms to protect against a wide range of hazards, often including items related to physical, noise or chemical protections. However, only 64% of workers wear the required equipment at all times [START_REF] Farooqui | Addressing the issue of compliance with personal protective equipment on construction worksites: A workers perspective[END_REF], either due to negligence or inconvenience.

II. PROPOSED APPROACH

Using domain knowledge to analyse a person's pose in relation to certain objects of interest, we can assess the conformal use of PPE by its proximity to certain body parts. By comparing the position of an equipment in question and the nearest most relevant body parts, it is possible to deduce whether it is properly worn through measurements in image space. The absolute scale of these measurements can be determined very accurately if needed by making use of the overlapping human pose, however, making use of relative measurements only is sufficient for the problem into consideration.

In our framework, object detection and pose estimation are performed by using the state-of-the-art deep neural networks YOLOV3 [START_REF] Redmon | Yolov3: An incremental improvement[END_REF] and OpenPose [START_REF] Cao | Openpose: Realtime multi-person 2d pose estimation using part affinity fields[END_REF] respectively. The former is used to localize various personal protective equipment in an image, the latter to localize the body limbs.

In detail, we explored two alternatives for the detection of hard hats, work shoes and harnesses. The first consists in assessing the degree of overlap between detected objects and relevant human joints (see II-A) while the second is based on asserting a number of conformal tests that need to be verified (see II-B).

A. Object -joints overlap

The bounding box associated to each detected object determines how many and which body parts are found in each box. This provides an overview of the relative position of each object with respect to one's posture, that is finely determined at the level of the head and the feet via OpenPose through a number of joints. We adopted this approach for the detection of hard hats and shoes, requiring that a sufficient number of relevant body limbs are located within the object's bounding box. Indicative results of hard hats and shoes detection are given in Fig. 1. This solution is only suitable if the equipment in question can overlap with limbs which can be obtained from pose estimation.

B. Harness and buckle detection

Some types of equipment either do not overlap with joints or the assessment of correct use goes beyond relative position estimation, as in the case of harnesses. In particular, a harness is considered properly worn if one of its buckle is located in the upper side of the torso. To introduce this constraint, we used 6 key points detected on the torso: The neck; left and right shoulders; left, mid and right hip. Checking whether the buckle is closer to neck than the middle hip would be sufficient, however, the detection of every body limb is not guaranteed.

The approach that we employed consists in asserting the validity of the following three conditions: (a) The buckle is closer to the neck than the middle hip; (b) The buckle is closer to the left shoulder than the right hip; (c) The buckle is closer to the right shoulder than the left hip. If the majority of these tests is positive, a buckle is deemed to be conformally worn. An example of harness and buckle detection is given in Fig. 1.

III. RESULTS

We evaluated the performance of joint object detection and human pose estimation. For this experiment, 418 pictures of people properly wearing hard hats, harness and shoes were collected.

The purpose of this test is to demonstrate the capacity of our implementation to detect whether the PPE are conformaly worn.

A qualitative result is provided in Fig. I. Both the YOLOV3 and Tiny-YOLOV3 with path aggregation networks [START_REF] Liu | Path aggregation network for instance segmentation[END_REF] configurations show high recalls.

The combination of human pose estimation and object detection enables the extraction of complex and robust in- formation in images. Such information would be difficult to obtain with object detection alone. However, pose estimation comes at a computing cost and increases the time required to handle an image. In the future, instance segmentation algorithms, such as Mask R-CNN [START_REF] He | Mask r-cnn[END_REF], could be used to extract finer details from images.

Fig. 1 .

 1 Fig. 1. Example of PPE detection in context using our approach.

Fig. 2 .

 2 Fig. 2. Example of various correctly worn equipment (green) and incorrectly worn (blue).

TABLE I FRAMEWORK

 I EVALUATION: COMPARISON BETWEEN TINY YOLOV3 AND YOLOV3 ON DETECTING WHETHER PPE ARE CONFORMALLY WORN.