Laurent Gaubert
email: gaubert@enib.fr

Pascal Redou

Jacques Tisseau

Tissular coupling and frequency locking I finite population

Introduction

Synchronization is an extremely important and interesting emergent property of complex systems. The first example found in literature goes back to the 17th century with Christiaan Huygens works [START_REF] Huygens | Horologium oscillatorium sive De motu pendulorum ad horologia aptato demonstrationes geometricae[END_REF][START_REF] Bennett | Huygens's clocks[END_REF]. This kind of emergent behavior can be found in artificial systems as well as in natural ones and at many scales (from cell to whole ecological systems). Biology abounds of periodic and synchronized phenomena and the works of Ilya Prigogine showed that such behaviors arise within specific conditions: a dissipative structure generally associated to a non-linear dynamic [START_REF] Prigogine | Introduction to Thermodynamics of Irreversible Processes[END_REF]. Biological systems are open, they evolve far from thermodynamic equilibrium and are subject to numerous regulating processes, leading to highly non-linear dynamics. Therefore periodic behaviors appear (with or without synchronization) at any scale [START_REF] Prigogine | Theoretical Physics and Biology[END_REF]. More generally, life itself is governed by circadian rhythms [START_REF] Gonze | Stochastic models for cicardian oscillations: Emergence of a biological rythm[END_REF]. Those phenomena are as much attractive as they are often spectacular: from cicada populations that appear spontaneously every ten or thirteen years [START_REF] Frank | Synchronization of periodical cicada emergences[END_REF] or networks of heart cells that beat together [START_REF] Donald | Mechanisms of sinoatrial pacemaker synchronization: A new hypothesis[END_REF] to huge swarms in which fireflies, gathered in a same tree, flash simultaneously. Furthermore, beyond biology one can find a wide source of examples in completely different fields of science (e.g. in behavioral psychology with the example of synchronizing applause [START_REF] Wiesenfeld | Synchronization transitions in a disordered josephson series array[END_REF]). For much more artificial and/or theoretical examples, one can consider the whole field of research that studies the coupling of smooth dynamical systems. Nowadays, it is one of the most important subject related to non-linear systems' dynamics, especially through the notion of chaotic systems' synchronization [START_REF] Pecora | Fundamentals of synchronization in chaotic systems, concepts, and applications[END_REF][START_REF] Goedgebuer | Laser cryptography by optical chaos[END_REF]. This wide source of examples leads the field of research to be highly interdisciplinary, from pure theory to concrete applications and experimentations. The classical concept of synchronization is related to the locking of the basic frequencies and instantaneous phases of regular oscillations. Those questions are usually addressed by studying specific kinds of coupled discrete or differential systems, using classical tools of the field (see for instance [START_REF] Pikovsky | Synchronization: A Universal Concept in Nonlinear Sciences[END_REF]). Convinced that synchronization phenomenon is completely natural in a large variety of coupled dynamical systems, we propose a new approach of the subject: firstly, we ask the question of synchronization differently than the usual way. Rather than trying to prove that synchronization actually takes place, we search conditions under which frequencies are locked as soon as the whole system oscillate. Secondly we enlarge the scope of handled models, by building a general framework for coupled systems called "tissular coupling". This framework is inspired by biological observations at cell's scale, but relevant at any scale of modeling. Under some general assumptions on the kind of interactions that constitute the coupling of the systems, we prove that for a wide class of tissular coupling systems, frequencies are mutually locked to a single value as soon as the whole population is oscillating.

This paper exhibits our model of tissular coupling and the frequency locking in the case of a finite number of coupled systems. In the first section we present some mathematical tools and the background we have used in order to study synchronization issue (the results exposed at the end of this paper is only a part of what we have fulfilled, and surely a really small part of what can be done using tissular coupling, this is why we state this framework in its general form). Then, we describe dynamical objects on which we focus, namely the tissular coupling and periodical motions of a population. In the second section we expose a useful way to reduce the problem to a structural one, with no more reference to the dynamics of the coupled systems. In the final section we exhibit some natural conditions under which we are able to prove the main result of this paper, a case of synchronization, in terms of frequencies locking. In a second paper [START_REF] Gaubert | Tissular coupling and frequency locking: II infinite population[END_REF] we expose the case of an infinite compact and connected population, which is processed with different mathematical tools.

Basic material and notation

As our model is inspired by cellular tissues, several terms clearly come from those kinds of complex systems. Nevertheless, its scope is not limited to cells nor cell tissues.

Model of population

Here are the basic compounds of our model:

• a population P is a measured space with finite measure. Typical populations are finite populations with counting measure or measurable subset of R m with Lebesgue measure. Every p ∈ P is called a cell; • for sake of concision, we suppose that the systems we want to study are R n -valued. Thus, a state of P is an element of S = M b (P, R n ), the space of measurable bounded applications from P to R n . A state of P will be denoted s, s(p) representing the state of the cell p;

• let I be a real interval and r ∈ N. A trajectory of P is an element of F(I, S). For sake of concision, any trajectory will be written s again, and s(t, p) stands for the state of cell p at time t (which should be written s(t)(p)). We then define the space of trajectories as:

T r = C r (I, S), ∀ compact J ⊂ I, sup J×P s(t, p) < ∞
A trajectory s is regular only along the time t (excepted in terms of measurability), thus we use the following unambiguous notation :

s ′ (t, p) = ∂ 1 s(t, p)
• as there is no ambiguity, all norms will be denoted in the same manner :

s ∈ S : s = sup P s(p) s ∈ T : s = sup I×P s(p, t) s(t, .) = sup P s(t, p) s(., p) = sup I s(t, p)
With this kind of norm, T 0 becomes obviously a Banach space.

Periodic motions and properties of periods on P Now, we define the notion of periodic trajectories on a population and some mathematical tools related to their study.

Definition 1. A period on P is a map τ from P to R * + . A trajectory s ∈ T r is said to be τ -periodic if for any p ∈ P, s(., p) is τ (p)-periodic. τ (p) is then called the period of p. The space of such trajectories is written T r τ For any s ∈ T 0 τ we define the τ -mean I τ (s) of s as:

I τ (s)(p) = 1 τ (p) τ (p) 0 s(t, p)dt Note that this quantity is R n -valued.
Here follows a corollary of the ergodic theorem [START_REF] Gaubert | Auto-organisation et émergence dans les systèmes couplés, individuation de données issues de systèmes biologiques couplés[END_REF] (and for references to ergodic theory see for instance [START_REF] Ya | Introduction to ergodic theory[END_REF][START_REF] Bunimovich | Dynamical Systems, Ergodic Theory and Applications[END_REF]):

Proposition 1 (Ergodic alternative). Let f a smooth real-valued map, τperiodic an τ ′ > 0. We define the τ ′ -average of f :

a τ ′ (f )(t) = lim N →+∞ 1 N + 1 N k=0 f (t + kτ ′ )
Then the following alternative holds:

• τ and τ ′ are Z-free (non commensurable), then a τ ′ (f ) is constant and equals τ I τ (f ); • there's an irreducible fraction a b such that bτ = aτ ′ , then

a τ ′ (f ) is τ ′ b - periodic.
The synchronization issues we want to study are related to the following question: as soon as the whole system oscillates, which assumptions insure that cells oscillate according to the same period. We will see that showing that periods are commensurable is the first natural step to achieve an answer to this question. We need the following notion: a (finite) subset {τ 1 , . . . , τ k } of R is said to be dependent (over Z) if there exists some integers l 1 , . . . , l k non all zeros and such that:

l 1 τ 1 + . . . + l k τ k = 0
Let's go back to differential systems on P now to complete our description of their coupling.

Differential systems on P

As we want to reach the most general result, each cell p is supposed to behave according to an autonomous differential system given by an application F p . Considering non-autonomous differential systems could be assimilated to the case of synchronization by an external force. Moreover, as we work with any kind of population, this external force could be assimilated to new cells and integrated in the population. For sake of concision, we will assume that F p is smooth. Then, we have a family of applications {F p } p∈P . We can define the extension F P of this family to S:

F P : S -→ S s -→ F P (s) with F P (s)(p) = F p (s(p))
The extension F P is said to be C r if every F p is C r and if for any bounded subset B of R n we have:

sup x∈B×p∈P dF p (x) < ∞
Coupler on P Now, we consider how to couple the differential systems given by F P . Usually [START_REF] Pikovsky | Synchronization: A Universal Concept in Nonlinear Sciences[END_REF], the coupling is made by adding to s ′ (t, p) a quantity that depends on s(t, .):

s ′ (t, p) = F p (s(t, p)) + c(s(t, .))(p) (1)
where c(s(t, .)), an element of S, is a function of the global state of P. This is what we naturally name a coupling in the final space. Another way of coupling is in the initial space :

s ′ (t, p) = F P c s(t, .) (p)
Coupling in the final space or in the initial space may have different interpretations. The first coupling method could be seen as physical exchanges and the second one as an instantaneous exchanges (for instance an high speed information exchange which could be modeled as instantaneous with regards to the system time scale). But, in most situations, those two couplings are equivalent. Indeed, the right-hand-side term of the equation ( 1) has the following shape: F (x) + ε so that, as soon as this r.h.s. stays in the range of the map F , one can find a vector y:

F (x) + ε = F (y)
writing δ = yx we have:

F (x) + ε = F (x + δ)
In fact, this case is quite general as ε is usually small, and even if it is not, we will study only periodical trajectories occurring essentially around limit cycles surrounded by the range of F . For these reasons, we will concentrate on the last kind of coupling. Now we give technical details about our method of coupling, by the way of a coupler :

C : S -→ S s -→ C(s)
This coupler is said to be linear if there is an application c satisfying:

c : R n × P -→ L (S, R n ) (x, p) -→ c(x, p)
such that for all p ∈ P:

C(s)(p) = c (s(p), p) .s
In other words, depending on the cell p (e.g. its spatial position) and on its state s(p), the coupling compute linearly (as a mean) the pseudo-state c (s(p), p) .s dictating the evolution of this cell, instead of the state s(p) which is used if the cell is isolated (without coupling). Sometimes the following point of view, the operator one, is be useful:

L c : S -→ L(S, S) s -→ L c (s)
where:

[L c (s).s](p) = c (s(p), p) .s
If L c is constant, the coupler is said to be uniformly linear and is naturally associated to an element of L(S, S). The case of uniformly linear coupler is the simpler one: the way a cell is influenced by its environment does not depends on its proper state. It is an approximation of what happens in real systems, but this simplification allows us to reach very global results.

To familiarize with the notion of coupler, let's write how it transforms a trajectory of P. If s ∈ T , the effect of a coupler on this trajectory is: C(s)(t, p) = c(s(t, p), p).s(t, .)

Tissular coupling

We are now able to define a tissular coupler. Given a population P, a coupler C is said to be a tissular coupler if the associated application c satisfies: c(x, p).s = P c d (x, p, q).s(q)dq + c a (x, p).s(p) where c d and c a are continuous and are called diffuse part and atomic part of c (and of C), respectively. The diffuse part represents the influence of the environment (the whole population) on the behavior of the cell p. The atomic part models the self-influence of p. We could have merge this into an unique linear operator, containing a Dirac impulse, but for both comprehension and manipulation purposes we think that this shape is preferable and less theoretical.

Finally, we can define a tissular coupling by specifying the equation that any solution should satisfy. Given a population P, a family of applications {F p } P and a tissular coupler C, the tissular coupling S(P, F P , C) is defined by the equation:

s ′ = F P • C(s) (2)
in other words, any solution s satisfies for all (t, p) ∈ I × P:

s ′ (t, p) = P c d (s(t, p), p, q).s(t, q)dq + c a (s(t, p), p).s(t, p)

We naturally end this introduction of the tissular coupling with a result on the existence of solutions, which is proved in a classical way [START_REF] Gaubert | Auto-organisation et émergence dans les systèmes couplés, individuation de données issues de systèmes biologiques couplés[END_REF]: Theorem 1. Suppose F P is C 1 and c d and c a are locally lipschtizian in their first variable. Given any initial condition (t 0 , s 0 ) in I ×S, the tissular coupling given by equation ( 2) admits a unique maximal solution.

We can then go further and begin to work on the heart of our matter.

Reduction of the synchronization problem

Cyclic application

The first reduction of the problem we have done was deciding to handle coupling in the initial space. Indeed, by the way of a natural assumption made on F P we will reduce the problem to a structural one. Definition 2. Let F be a map from R n to R n and U an open subset of R n . F is said to be cyclic on U if for any motion x : R → U such that there exists τ satisfying:

F • x(t + τ ) = F • x(t) ∀t ∈ R then: x(t + τ ) = x(t) ∀t ∈ R
Before we expose what the problem of frequency locking becomes under this assumption, we should explain why considering cyclic maps is somewhat natural. For example, if one tries to build an elementary non-cyclic map, one can consider the following:

F : R 2 -{0} → R 2 -{0} x = x 1 x 2 → 1 x -x 2 x 1
for which any map x(t) satisfying :

x(t) = r(t) cos(t) r(t) sin(t)
leads to (fig. 1):

F • x(t + ω) = F • x(t)
But this kind of applications is not of great interest in the scope of physical and biological phenomena, indeed if we want to consider a little more credible version of this map, we must look at the mathematical pendulum, which is defined by:

F : R 2 -{0} → R 2 -{0} x = x 1 x 2 → -x 2 x 1
for which the preceding property does not hold any more (it's a one-to-one map). Moreover, even if the map isn't one-to-one, in a realistic model of a physical phenomenon, periodic trajectories are rare: given such a dynamical system, any specific periodic trajectory in the phase space corresponds to a limit cycle. Those limit cycles are indeed rare in non conservative systems (compare for instance the mathematical and the physical pendulum). In other words, there exists a neighborhood of those limit cycles in which the only maps x(t) such that F • x(t) is periodic are precisely the limit cycles themselves, which are effectively periodic applications. Moreover, this condition of cyclicity is extremely useful and allows us to prove the following result:

Proposition 2. Let (P, C, F P ) be a tissular coupling and τ a period on P.

Let's assume that there exists a family {U p } p∈P of open subsets of R n such that for all p ∈ P F p is cyclic on U p . If s is a τ -periodic solution of the coupling such that for all p ∈ P, s(I, p) ⊂ U p , then s belongs to the following set:

A(τ, C) = T ∞ ∪ C -1 (T ∞ )
We then have drastically reduced the problem: under hypotheses of the proposition 2 we can throw F P and only work with the coupler C and the space of periodic solutions. This last section exhibits the main results of our work on finite populations.

Synchronization in finite population

Here we focus on a coupler in the case of a finite population P:

P = {p 1 , . . . , p k }
If C is a tissular coupling, the diffuse part is sufficient to define it, thus we can also write it for x ∈ R n as:

c d (x, p i , pj) = c ij (x) ∈ L (R n )
L c adopts naturally a matrix shape:

L c (x) =    c 11 (x) . . . c 1k (x) . . . . . . . . . c k1 (x) . . . c kk (x)   
We point out that each c ij (x) is a linear application, an element of L (R n , R n ) (to be precise and to anticipate on some theoretical generalizations of the tissular coupling, we note that L c (x) is the matrix of a linear application on (R n ) k seen as a module on the ring

L (R n , R n )).
The image of the motion s(., p i ) is thus given by:

C(s)(t, p i ) = k j=1 c ij (s(t, p i ))s(t, p j )
The kind of assumptions we are about to make on C involve linear properties.

Especially we work with the rank of L c and of sub-matrices. Thus, if J ⊂ {1, • • • , k} contains l elements, we write

I c = {1, • • • , k} -J. If M = (m ij
) is an k × k matrix, we define M I as the l × (lk) matrix which coefficients are the m i,j for (i, j) ∈ I × I c . For s ∈ S or T we write s I the vector [s(p i )] i∈I or [s(., p i )] i∈I (see figure 2).

(R n , R n )). C is non-reducible if it's not reducible.
The next step consists in "cleaning" the space of periodic solutions in order to exclude some degenerated and/or trivial cases. First the trivial periodic solutions: Definition 4. Any s in A(τ, C) is said to be non trivial if for all p ∈ P, neither s(., p) and c(e)(., p) are constant maps. Now we consider some trajectories that are degenerated in a stronger sense than the one of trivial trajectories: our aim is to avoid periodic trajectories which oscillating behaviors are the only consequences of the coupler's dynamic (we focus on the synchronization of interacting oscillating systems, not on some systems oscillating due to an external force, and we have already mentioned that this kind of external force can be included in the population itself). Such trajectories are those for which there exists a state b satisfying:

L c s(t, .) s(t, .) -b = 0
In terms of matrices, we then have for the cell p i :

k j=1 c ij s(t, p i ) s(t, p j ) -b(p j ) = 0
In other words, the preceding equality shows that the vector [s(., p 1 ), • • • , s(., p k )] is not linearly free, according to the compounds of L c e(t, .) . These observations lead us to the following definition: Definition 5. Let P be a finite population, C a tissular coupler on P and τ a period on P. A non-trivial element s ∈ A (τ, C) is said to be C-free if for any b ∈ S and for any non-trivial J ⊂ {1, • • • , k} we have:

L c s(t, .) J . s(t, .) -b J c = 0
In fact, we can prove that, when C is non-reducible, a non C-free element is degenerated because it comes from a solution of a system on a population which cardinal is strictly lower than k.

Here is the first step to frequencies locking as it deals with dependency of periods [START_REF] Gaubert | Auto-organisation et émergence dans les systèmes couplés, individuation de données issues de systèmes biologiques couplés[END_REF]: Theorem 2. If C is non-reducible and if A (τ, C) contains a non-trivial and C-free element, then τ (P) is a dependent set.

Finally we extend this result to a more powerful conclusion. In order to prove synchronization, we must add a last condition on C. This is done introducing the following definition: Definition 6. Let C be a uniformly linear coupler on P. C is said to be strongly non-reducible if it is non-reducible and if for all x ∈ R n and all non trivial subset J ⊂ {1, . . . , k} with more than k 2 elements, L J c (x) is injective.

Remark 1. Because of its dimensions, L J c (x) can't be injective if J contains less than k 2 elements.

We can now state our main result [START_REF] Gaubert | Auto-organisation et émergence dans les systèmes couplés, individuation de données issues de systèmes biologiques couplés[END_REF]:

Theorem 3. Let P be a finite population of cardinal k, C a tissular coupler, uniformly linear and strongly non-reducible and τ a period on P. Suppose s ∈ A (τ, C) isn't trivial and is non-reducible. If more than k 2 cells have the same period then all cells have the same period.

Conclusion

In this work we have built a general framework, the tissular coupling, to handle a wide variety of coupled systems, and therefore a wide class of complex systems. We focused on an emergent property of those dynamical systems: synchronization, and precisely frequencies locking. We used the notion of tissular coupling to show that the synchronization issue may be addressed differently. Usually one observes solutions of particular coupled systems and shows that within suitable conditions synchronization must occur. Those results are often qualitatively dependent of the studied systems and do not stand in the general case. We tried to change our point of view and to bring completing results. These are less precise than usual ones as we don't prove that synchronization ultimately happens. Instead, we consider the problem at its end: if one supposes that coupled systems oscillate, then they must be synchronized. The loss in time evolution informations is compensated by very general results, almost independent from the individual differential systems to be coupled. We believe that this complementary approach of frequencies locking will lead to future developments as it brings about many unused mathematical tools. This will be showed in a following paper in which we study the same question concerning an infinite compact connected populations but with very different methods [START_REF] Gaubert | Tissular coupling and frequency locking: II infinite population[END_REF]. Other examples of the use of tissular coupling can be found in [START_REF] Gaubert | Auto-organisation et émergence dans les systèmes couplés, individuation de données issues de systèmes biologiques couplés[END_REF] on emergence of spatial patterns. All those work and results are stimulating and seem to prove that this framework will be fruitful.

Fig. 1 .

 1 Fig. 1. Vector field associated to F and an example of non periodic map x(t) such that F • x is periodic.

Fig. 2 .Definition 3 .

 23 Fig. 2. Matrix and vectors associated to a subset J of {1, • • • , k}.


Rejeb Hadiji
email: hadiji@u-pec.fr

Habib Yazidi
email: habib.yazidi@u-pec.fr

A nonlinear general Neumann problem involving two critical exponents

Keywords: Sobolev critical exponent, The trace embedding, Variational problem, Critical nonlinearity in the boundary, Palais-Smale Condition, The mean curvature. 2010 AMS subject classifications: 35J20, 35J25, 35J60

We discuss the existence of solutions to the following nonlinear problem involving two critical Sobolev exponents

where β ≥ 0, Q is continuous on ∂Ω, p ∈ H 1 (Ω) is continuous and positive in Ω and f is a lower-order perturbation of |u| 2 * -1 with f (x, 0) = 0.

Introduction

In this work, we deal with the following problem

     -div(p(x)∇u) = β|u| 2 * -2 u + f (x, u) in Ω, u ≡ 0 in Ω, ∂u ∂ν = Q(x)|u| 2 * -2 u on ∂Ω, (1.1)
where Ω ⊂ IR N , N ≥ 3, is a bounded domain with the smooth boundary ∂Ω, ν is the outer normal on ∂Ω, β ≥ 0 is a constant, the coefficient Q is continuous on ∂Ω, the coefficient p ∈ H 1 (Ω) is continuous and positive in Ω and f (x, u) : Ω × IR → IR is measurable in x, continuous in u.

Here, 2 * = 2(N -1) N -2 is the critical Sobolev exponent for the trace embedding of the space H 1 (Ω) into L 2 * (∂Ω) and 2 * = 2N N -2 is the critical Sobolev exponent for the embedding H 1 (Ω) into L 2 * (Ω). Both embedding are continuous, but not compact. Our goal is to study the existence of solutions to problem (1.1). The main motivation to consider such problem is the study of conformal deformations of Riemannian manifolds with boundary, see [START_REF] Cherrier | Problèmes de Neumann non linéaires sur les variétés Riemanniennes[END_REF], [START_REF] Escobar | Sharp constant in a Sobolev trace inequality[END_REF] and [START_REF] Escobar | Uniqueness theorems on conformal deformation of Riemannian metric to a scalar flat metric with constant mean curvature[END_REF]. Problem(1.1) has a variational form. Then the eventual solutions correspond to the critical points of the energy functional.

The existence of a solution of (1.1) is closely related to S (resp. S 1 ) which is the best Sobolev constant for the imbedding H 1 (Ω) into L 2 * (Ω) (resp. for the imbedding H 1 (Ω) into L 2 * (∂Ω)). As in [START_REF] Brezis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF] for the nonlinear Dirichlet problem with critical Sobolev exponent, we will fill out the sufficient conditions to find solutions for the problem in presence of a nonlinear Neumann boundary data with a critical nonlinearity. One of the difficulty of our problem, besides the fact that the associated functional does not satisfy the Palais-Smale compactness condition (PS), is that it possesses four levels of homogeneity.

Let us recall some works related to the problem (1.1). If p ≡ 1 and u satisfies homogeneous Dirichlet condition, problem (1.1) has been treated in [START_REF] Brezis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF], where the authors obtained positive solutions with energy less than 1 N S N 2 , see also [START_REF] Hadiji | Solutions positives de l'équation -∆u = u p + µu q dans un domaine à trou[END_REF] and [START_REF] Crouau | Critical Sobolev exponent and the dimension three[END_REF]. In [START_REF] Struwe | A global compactness result for elliptic boundary value problems involving limiting nonlinearities[END_REF], the author gives a complete description of the energy levels c, associated to problem (1.1), on which (P S) c sequence is not compact. For the case p ≡ 1, f (x, u) = λ u with homogeneous Dirichlet condition we refer the reader to [START_REF] Hadiji | Problem with critical Sobolev exponent and with weight[END_REF][START_REF] Hadiji | Localization of solutions for nonlinear elliptic problems with critical growth[END_REF] . For the homogeneous Neumann problem, in [START_REF] Chabrowski | On the critical Neumann problem with perturbations of lower order[END_REF], the authors proved the existence of solution with energy less than 1 2N S N 2 . The case p ≡ 1 ≡ Q, β = 0 and f (x, u) is a linear perturbation, has an extensive literature and the first existence results was treated in [START_REF] Adimurthi | The Neumann problem for elliptic equation with critical non linearity, 65th birthday of Prof. Prodi[END_REF][START_REF] Adimurthi | Positive solution for Neumann problem with critical non linearity on boundary[END_REF][START_REF] Chabrowski | Least energy solutions of a critical Neumann problem with weight[END_REF][START_REF] Chabrowski | Sharp Sobolev Inequality Involving a Critical Nonlinearity on a Boundary[END_REF]. In this case the solutions are obtained as minimizers of the variational problem associated to (1.1) with energy less than S 1 . If β = 0 and f (x, u) has an explicit form, problem (1.1) has been studied in [START_REF] Yazidi | On some nonlinear Neumann problem with weight and with critical Sobolev trace maps[END_REF][START_REF] Yazidi | On nonhomogeneneous Neumann problem with weight and with critical nonlinearity in the boundary[END_REF] and some existence results are obtained.

In [START_REF] Chabrowski | Positive solutions of a Neumann problem with competing critical nonlinearities[END_REF], the authors were interested to the case p ≡ 1, f (x, u) = 0 and the presence of two critical nonlinearities. They derived some existence results by the use of the concentration compactness principle see [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The limit case, Part 1[END_REF]. For another form of equation (1.1) with competing critical nonlinearities, see [START_REF] Pierotti | On a Neumann problem involving two critical exponents: remarks on geometrical and topological aspects[END_REF] and references therein.

In this paper we are concerned with the general case, more precisely, p ≡ 1, Q ≡ 0 and f (x, u) = 0. We assume that f is a lower-order perturbation of |u| 2 * -1 and f (x, 0) = 0.

Let

p 0 = min x∈ Ω p(x) and x 0 ∈ ∂Ω satisfy (Q(x 0 )) N -2 p(x 0 ) = max x∈∂Ω |Q(x)| N -2 p(x) .
We assume that

(1.2) |p(x) -p(x 0 )| = o(|x -x 0 |) and (1.3) |Q(x) -Q(x 0 )| = o(|x -x 0 |) for x near x 0 .
Our first contribution to problem (1.1), in section 2, is an existence result for the case where β = 0. The energy solutions which we find are under the level on which the (PS) condition failed. More precisely, we show existence of solutions with energy in ]0,

1 2(N -1) p(x 0 ) (Q(x 0 )) N-2 S N -1 1 [.
Next, in section 3, we turn to the general case and look for solutions for problem (1.1) in the case of the presence of competing critical nonlinearities in the case p(x 0 ) = p 0 . The main difficulty of the problem in caused by the presence of two critical exponents and a general nonlinear perturbation. This fact causes the change in energy level for which the Palais Smale condition (PS) is not satisfied. In this paper, we determine explicitly the new energy level M (S, S 1 ) defined by

(1.4) M (S, S 1 ) = p(x 0 ) (Q(x 0 )) N-2 S N -1 1 2 N -2 1 + √ 1 + 4E N -2 1 N - N -2 N (N -1) 1 1 + √ 1 + 4E where E = p(x 0 ) Q(x 0 ) N-2 S N-1 1 (p 0 S) N 2 2 N-2
. We will show the existence of solution for (1.1) with energy in ]0, M (S, S 1 )[. Note that 0 < M (S, S 1 ) < min 1 2(N -1)

p(x 0 ) (Q(x 0 )) N -2 S N -1 1 , 1 N (p 0 S) N 2
.

2 Existence results for β = 0

We assume that f (x, u) can be written as

(2.1) f (x, u) = a(x)u + g(x, u), with (2.2) a(x) ∈ L ∞ (Ω), (2.3) there exists 2 < α ≤ 2 * such that, for every x ∈ IR N and u ∈ IR, αG(x, u) ≤ u g(x, u), where G(x, u) = u 0 g(x, t)dt, (2.4) |g(x, u)| = o(|u|) as u → 0, uniformly in x, (2.5) |g(x, u)| = O(|u| 2 * -1 ) as |u| → +∞, uniformly in x. or (2.6) |g(x, u)| = O(|u| r-1 ) as |u| → +∞, uniformly in x,
where r is such that 2 * < r < 2 * , Moreover, we assume that the first eigenvalue λ 1 (a) of the following problem is positive:

-div(p(x)u) -a(x)u = µu in Ω ∂u ∂ν = 0 on ∂Ω, That is, (2.7) λ 1 (a) = inf u ∈H 1 (Ω) Ω |∇u| 2 -a(x)u 2 dx, Ω u 2 dx = 1 > 0.
Under assumption (2.7), it is easy to verify that ||u||

= ( Ω |∇u| 2 -a(x)u 2 dx) 1 2 is a norm on H 1 (Ω) equivalent to the usual norm . H 1 . Let Φ(u) = 1 2 Ω p(x)|∇u| 2 dx - Ω F (x, u)dx - 1 2 * ∂Ω p(x)Q(x)|u| 2 * ds x , u ∈ H 1 (Ω),
where

F (x, u) = u 0 f (x, t)dt for x ∈ Ω, u ∈ IR.
v 0 ∈ H 1 , v 0 ≥ 0 on Ω, v 0 = 0 on ∂Ω, such that sup t≥0 Φ(tv 0 ) < 1 2(N -1) p(x 0 ) (Q(x 0 )) N -2 S N -1 1 .
Then problem (1.1) possesses a solution.

Proof of Theorem 2.1 Let s = 2 * when f satisfies (2.5) and s = r when f satisfies (2.6). By (2.4) we have, for any ε > 0, there is a δ > 0 such that |g(x, u)| ≤ ε|u| for a.e x ∈ Ω, and for all |u| ≤ δ, thus, by (2.5) or (2.6), we obtain |g(x, u)| ≤ ε|u| + C|u| s-1 for a.e x ∈ Ω, and for all u ∈ IR, and for some constant C (depending on ε). Therefore, we have (2.9)

F (x, u) ≤ 1 2 a(x)u 2 + ε 2 u 2 + C s
|u| s for a.e x ∈ Ω, and for all u ∈ IR.

Hence we find, for all u ∈ H 1 (Ω),

Φ(u) ≥ 1 2 Ω p(x)|∇u| 2 dx- 1 2 Ω a(x)|u| 2 dx- ε 2 Ω |u| 2 dx- C s Ω |u| s dx- 1 2 * ∂Ω p(x)Q(x)|u| 2 * ds x
Using (2.7) we easily see that, for ε > 0 small enough , there exist constants k > 0,

C 1 > 0 and C 2 > 0 such that Φ(u) ≥ k u 2 -C 1 u s -C 2 u 2 * ≥ u 2 k -C 1 u s-2 -C 2 u 2 * -2 for all u ∈ H 1 ,
which implies, since 2 * > 2 and s > 2, for some small α > 0 there exists ρ > 0 such that (2.10) Φ(u) ≥ ρ, provided u = α.

At this stage, we need some notations and some estimations. We recall S 1 defined by

S 1 = inf IR N + |∇u| 2 dx; u ∈ H 1 (IR N + ), IR N-1 |u| 2 * dx = 1
the best constant for the trace embedding

H 1 (IR N + ) into L q (∂IR N + ), where R N + = {x = (x ′ , x N ) : x ′ ∈ IR N -1 , x N > 0}.
We recall from [START_REF] Escobar | Sharp constant in a Sobolev trace inequality[END_REF] and [START_REF] Lions | The concentration-compactness principle in the calculus of variations. The limit case, Part 1[END_REF] that the minimizing functions of S 1 are of the form

(2.11) W (x) = γ N |x ′ | 2 + (1 + x N ) 2 N-2 2
, where γ N is a positive constant depending on N . We set

W ε,x 0 (x) = ε -N-2 2 φ(x)W ( x -x 0 ε ),
where

x 0 ∈ ∂Ω and φ is a radial C ∞ -function such that φ(x) = 1 if |x -x 0 | ≤ R 4 0 if |x -x 0 | > R 2
with R > 0 is a small constant.

From [START_REF] Adimurthi | Positive solution for Neumann problem with critical non linearity on boundary[END_REF] and [START_REF] Chabrowski | Sharp Sobolev Inequality Involving a Critical Nonlinearity on a Boundary[END_REF] we have the following estimates (2.12)

Ω p(x)|∇W ε,x 0 | 2 dx = p(x 0 )A 1 -p(x 0 )H(x 0 )    A ′ 2 ε| log ε| + o(ε| log ε|) if N = 3 A 2 ε + o(ε) if N ≥ 4, (2.13)
∂Ω p(x)Q(x)|W ε,x 0 | 2 * ds x = p(x 0 )Q(x 0 )(B 1 -H(x 0 )B 2 ε) + o(ε)
where A 1 , A ′ 2 , A 2 , B 1 and B 2 are some positive constants defined explicitly in [START_REF] Adimurthi | Positive solution for Neumann problem with critical non linearity on boundary[END_REF]. From [START_REF] Yazidi | On some nonlinear Neumann problem with weight and with critical Sobolev trace maps[END_REF], for some 2 < r < 2 * , we have

(2.14) Ω |W ε,x 0 | r dx =    o(ε) if N ≥ 4 o(ε| ln(ε)|) If N = 3.
Let us notice that (2.15)

S 1 = A 1 B 2 2 * 1 and A 2 - 2 2 * A 1 B 2 B 1 > 0.
On the other hand, when f satisfies (2.5), we easily see that lim t→+∞ Φ(tW ε,x 0 ) = -∞.

Then we take v = t 0 W ε,x 0 , where t 0 > 0 is chosen large enough so that v > α and Φ(v) ≤ 0. When f satisfies (2.6), using (2.12)-(2.14), we have

Φ(tW ε,x 0 ) = t 2 A -t 2 * B + t r    o(ε) if N ≥ 4 o(ε| ln(ε)|) if N = 3.
Therefore, for ε > 0 small enough, there exists many t 0 > 0 such that t 2 0 At 2 * 0 B < 0. Let, again, v = t 0 W ε,x 0 for ε small enough when t 0 is chosen large such that v > α and Φ(v) ≤ 0. Set (2.16)

c = inf P∈A max w∈P Φ(w),
where A denotes the class of continuous paths joining 0 to v. Thanks to a result of Ambrosetti and Rabinowtz [START_REF] Ambrosetti | Dual variational methods in critical point theory and applications[END_REF], see also [START_REF] Brezis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF], there exists a sequence

{u j } in H 1 (Ω) such that Φ(u j ) → c and Φ ′ (u j ) → 0 in H -1 (Ω).
Looking at (2.8) we see that c <

1 2(N -1) p(x 0 ) (Q(x 0 )) N-2 S N -1 1 .
In order to conclude the proof of Theorem 2.1, we need the following Lemma.

Lemma 2.1 Let {u j } ⊂ H 1 (Ω) be a sequence satisfying (2.17) Φ(u j ) → c < p(x 0 )S N -1 1 2(N -1)(Q(x 0 )) N -2 and (2.18) Φ ′ (u j ) → 0 in H -1 (Ω) then {u j } is relatively compact in H 1 (Ω).

Proof of Lemma 2.1:

We start by showing that {u j } is bounded in H 1 (Ω). Using (2.1) and (2.7) we see that (2.17) and (2.18) are equivalent to

(2.19) 1 2 u j 2 - Ω G(x, u j )dx - 1 2 * ∂Ω p(x)Q(x)|u j | 2 * ds x = c + o(1),

and

(2.20)

u j 2 - Ω g(x, u j )u j dx - ∂Ω p(x)Q(x)|u j | 2 * ds x =< ξ j , u j > with ξ j → 0 in H -1 . Taking (2.19)-1 2 (2.20), we get (2.21) 1 2(N -1) ∂Ω p(x)Q(x)|u j | 2 * ds x - Ω G(x, u j )dx + 1 2 Ω g(x, u j )u j dx = c + o( u j ).
On the other hand, (2.19)-1 2 * (2.20) yields

(2.22) 1 2(N -1) u j 2 - Ω G(x, u j )dx + 1 2 * Ω g(x, u j )u j dx = c + o( u j ).
Using (2.3), (2.21) and (2.22) follow

(2.23) 1 2(N -1) u j 2 -(1 - α 2 * ) Ω G(x, u j )dx ≤ c + o( u j ) and (2.24) 1 2(N -1) ∂Ω p(x)Q(x)|u j | 2 * ds x -(1 - α 2 ) Ω G(x, u j )dx ≤ c + o( u j ). Computing ( α 2 -1)(2.23)+(1 -α 2 * )(2.24), we obtain ( α 2 -1) u j 2 + (1 - α 2 * ) ∂Ω p(x)Q(x)|u j | 2 * ds x ≤ c + o( u j ).
Therefore, since 2 < α ≤ 2 * , we obtain that {u j } is bounded in H 1 (Ω). Extract a subsequence, still denoted by u j , such that

u j ⇀ u weakly in H 1 (Ω), u j → u strongly in L t (Ω) for all t < 2 * = 2N N -2 , u j → u a.e. on Ω, f (x, u j ) → f (x, u) strongly in L r r-1 (Ω),
u j ⇀ u weakly in L 2 * (∂Ω).

Passing to the limit in (2.18), we obtain

-div(p(x)∇u) = f (x, u) in Ω ∂u ∂ν = Q(x)|u| 2 * -2 u on ∂Ω
We shall now verify that u ≡ 0. Indeed , suppose that u ≡ 0. We claim that Therefore

Ω f (x, u j )u j dx → 0 and Ω F (x, u j )dx → 0. From (2.5) or (2.6), let s = 2 * if f satisfies (2.5) and s = r if f satisfies (2.6), we have for some constants C 1 > 0 and C 2 > 0 |f (x, u)| ≤ C 1 |u| s-1 + C 2
Ω f (x, u j )u j dx ≤ C 1 Ω |u j | s dx + C 2 Ω |u j |dx and Ω F (x, u j )dx ≤ C 1 s Ω |u j | s dx + C 2 Ω |u j |dx.
Since u j → 0 in L s (Ω) then for j large enough, we have

Ω f (x, u j )u j dx = o(1)
and

Ω F (x, u j )dx = o(1).
Which gives the desired result. Extracting a subsequence, still denoted by u j , we may assume that (2.25)

Ω p(x)|∇u j | 2 dx → l for some constant l ≥ 0.
Passing to the limit in (2.20), we obtain

(2.26) ∂Ω p(x)Q(x)|u j | 2 * ds x → l.
Passing to the limit in (2.21), we easily get

(2.27) 1 2(N -1) l = c.
Therefore l > 0 and ∂Ω p(x)Q(x)|u j | 2 * ds x > 0 for large j.

On the other hand, from the result of [24, Theorem 02], we know that there exists a constant C(Ω) > 0 such that for every w ∈ H 1 (Ω)

Ω |∇w| 2 dx + C(Ω) Ω |w| k dx ≥ S 1 ∂Ω |w| 2 * ds x 2 2 * , with k = 2N N -1 if N ≥ 4 and k > 3 = 2N N -1 if N = 3.
We apply this result for w j = (p(x)) 1 2 u j and in particular for N = 3 we take k such that 6 = 2N N -2 > k > 3, we obtain for j large enough

Ω |∇(p(x)) 1 2 u j | 2 dx + C(Ω) Ω |(p(x)) 1 2 u j | k dx ≥ S 1 ∂Ω |(p(x)) 1 2 u j | 2 * ds x 2 2 * Since k < 2N
N -2 for every N ≥ 3, thanks to the compact embedding H 1 (Ω) ֒→ L k (Ω), we have, for a subsequence, u j → 0 strongly in L k (Ω) and we deduce (2.28)

Ω p(x)|∇u j | 2 dx + o(1) ≥ S 1 ∂Ω |(p(x)) 1 2 u j | 2 * ds x 2 2 * + o(1). Using the fact that |Q(x)| N -2 p(x) ≤ (Q(x 0 )) N -2 p(x 0 ) ∀x ∈ ∂Ω, (2.28) becomes (2.29) Ω p(x)|∇u j | 2 dx + o(1) ≥ S 1    ∂Ω (p(x)) 2 * 2   |Q(x)| N-2 p(x) (Q(x 0 )) N-2 p(x 0 )   1 N-2 |u j | 2 * ds x    2 2 * + o(1) ≥ S 1 (p(x 0 )) 1 N-2 Q(x 0 ) 2 2 * ∂Ω p(x)|Q(x)||u j | 2 * ds x 2 2 * + o(1) ≥ S 1 (p(x 0 )) 1 N-2 Q(x 0 ) 2 2 * ∂Ω p(x)Q(x)|u j | 2 * ds x 2 2 * + o(1).
At the limit we obtain

l ≥ (p(x 0 )) 1 N-2 Q(x 0 ) 2 2 * S 1 l N-2 N-1 and l ≥ (p(x 0 )) 1 N-1 (Q(x 0 )) N-2 N-1 S 1 l N-2 N-1 .
Using (3.9) and (2.27) we see that l ≡ 0 and

l 1 N-1 ≥ (p(x 0 )) 1 N-1 (Q(x 0 )) N-2 N-1 S 1 . Therefore l ≥ p(x 0 ) (Q(x 0 )) N -2 S N -1 1 and from (2.27) we have c ≥ 1 2(N -1) p(x 0 ) (Q(x 0 )) N -2 S N -1 1
which gives a contradiction with the fact that c <

1 2(N -1) p(x 0 ) (Q(x 0 )) N-2 S N -1 1
, thus u ≡ 0. Now, we shall prove, for a subsequence, that u j → u strongly in H 1 (Ω). We start by showing that Φ(u) ≥ 0. Indeed, since u is a solution of (1.1) with β = 0, we have

Ω p(x)|∇u| 2 dx = Ω f (x, u)dx + ∂Ω p(x)Q(x)|u| 2 * ds x .
On the other hand

Φ(u) = 1 2 Ω p(x)|∇u| 2 dx - 1 2 * ∂Ω p(x)Q(x)|u| 2 * ds x - Ω F (x, u)dx.
Therefore, using (2.3), we have

Φ(u) ≥ 1 2(N -1) Ω p(x)|∇u| 2 dx + ( α 2 * -1) Ω F (x, u)dx, and Φ(u) ≥ 1 2(N -1) ∂Ω p(x)Q(x)|u| 2 * ds x + ( α 2 -1) Ω F (x, u)dx.
Since 2 < α ≤ 2 * , we deduce that φ(u) ≥ 0. We set v j = u ju. We have (2.30)

Ω p(x)|∇u j | 2 dx = Ω p(x)|∇u| 2 dx + Ω p(x)|∇v j | 2 dx + o(1)
and from [START_REF] Brezis | A relation between pointwise convergence of functions and convergence of functionals[END_REF] we deduce that (2.31)

∂Ω p(x)Q(x)|u j | 2 * ds x = ∂Ω p(x)Q(x)|u| 2 * ds x + ∂Ω p(x)Q(x)|v j | 2 * ds x + o(1
Φ(u) + 1 2 Ω p(x)|∇v j | 2 dx - 1 2 * ∂Ω p(x)Q(x)|v j | 2 * ds x = c + o(1)
and (looking at (2.18))

(2.33)

Ω p(x)|∇v j | 2 dx - ∂Ω p(x)Q(x)|v j | 2 * ds x = o(1).
Extracting a subsequence, still denoted by u j , we may assume that

Ω p(x)|∇v j | 2 dx → l for some constant l ≥ 0.
From (2.33) we obtain

∂Ω p(x)Q(x)|v j | 2 ds x = l.
Passing to the limit in (2.32), we easily see that

(2.34) 1 2(N -1) l = c -Φ(u).
Using the Sobolev embedding, see (2.29) for details, we have

(2.35) l ≥ (p(x 0 )) 1 N-2 (Q(x 0 )) N-2 N-1 S 1 l N-2 N-1 .
We claim that l = 0. Indeed, arguing by contradiction, assuming that l = 0, then (2.35) gives

l ≥ p(x 0 ) (Q(x 0 )) N -2 S N -1 1 . From (2.34), we obtain c -Φ(u) ≥ 1 2(N -1) p(x 0 ) (Q(x 0 )) N -2 S N -1 1 which gives a contradiction, since c < 1 2(N -1) (p(x 0 ) (Q(x 0 )) N-2 S N -1 1
and Φ(u) ≥ 0. Therefore l = 0, c = Φ(u) and u j → u strongly in H 1 (Ω).

2.1 Sufficient conditions on f (x, u) which give condition (2.8):

We claim that W ε,x 0 satisfies condition (2.8) for ε > 0 sufficiently small. Indeed, we have

Φ(tW ε,x 0 ) = 1 2 t 2 Ω p(x)|∇W ε,x 0 | 2 dx - t 2 * 2 * ∂Ω p(x)Q(x)|W ε,x 0 | 2 * ds x - Ω F (x, tW ε,x 0 )dx.
When f satisfies (2.5), we easily see that lim t→+∞ Φ(tW ε,x 0 ) = -∞ and for large t 0 > 0 we have Φ(t 0 W ε,x 0 ) < 0. When f satisfies (2.6), using (2.12), (2.13) and (2.14), we have

Φ(tW ε,x 0 ) = t 2 A -t 2 * B + t r    o(ε) if N ≥ 4 o(ε| ln(ε)|) if N = 3.
Therefore, for ε > 0 small enough, we chose t 0 > 0 such that t 2 0 At 2 * 0 B < 0 and Φ(t 0 W ε,x 0 ) < 0. Therefore, in both cases, sup t∈[0, 1] Φ(t t 0 W ε,x 0 ) is achieved at some 0 ≤ tε ≤ 1 and tε is bounded. In the rest of this section, we note t ε = tε t 0 . From now, we can suppose that t ε > 0, indeed if t ε = 0 then sup t≥0 Φ(tW ε,x 0 ) = 0 and the condition (2.8) is satisfied. Since the derivative of the function t → Φ(tW ε,x 0 ) vanishes at t ε we have (2.36)

t ε Ω p(x)|∇W ε,x 0 | 2 dx -t 2 * -1 ε ∂Ω p(x)Q(x)|W ε,x 0 | 2 * ds x - Ω f (x, t ε W ε,x 0 )W ε,x 0 dx = 0.
We claim that (2.37)

Ω f (x, t ε W ε,x 0 )W ε,x 0 t ε dx → 0 as ε → 0.
Indeed, from (2.36), we have

Ω p(x)|∇W ε,x 0 | 2 dx -t 2 * -2 ε ∂Ω p(x)Q(x)|W ε,x 0 | 2 * ds x - Ω f (x, t ε W ε,x 0 )W ε,x 0 t ε dx = 0.
Using (2.1)-(2.5) or (2.1)-(2.4) and (2.6), there are C 1 > 0 and C 2 > 0 such that, for a.e.

x ∈ Ω, for all u ∈ IR, |f

(x, u)| ≤ C 1 |u| s-1 + C 2 |u| where s = 2 * if f satisfies (2.5) and s = r if f satisfies (2.6). Therefore Ω f (x, t ε W ε,x 0 )W ε,x 0 t ε dx ≤ C 1 t ε s-2 W ε,x 0 s L s + C 2 W ε,x 0 2 L 2 ,
Using the fact that, as ε → 0, t ε is bounded, W ε,x 0 L 2 (Ω) → 0 and W ε,x 0 L s (Ω) → 0 since s < 2 * , we get directly (2.37). Consequently, for ε > 0 small enough, (2.36) become

t ε Ω p(x)|∇W ε,x 0 | 2 dx -t 2 * -1 ε ∂Ω p(x)Q(x)|W ε,x 0 | 2 * ds x = o(1). Therefore (2.38) t ε ≤     Ω p(x)|∇W ε,x 0 | 2 dx ∂Ω p(x)Q(x)|W ε,x 0 | 2 * ds x     1 2 * -2 + o(1). Set X ε =     Ω p(x)|∇W ε,x 0 | 2 dx ∂Ω p(x)Q(x)|W ε,x 0 | 2 * ds x     1 2 * -2 and M ε = sup t∈[0, 1] Φ(t t 0 W ε,x 0 ) = Φ(t ε W ε,x 0 ). Since the function t → 1 2 t 2 Ω p(x)|∇W ε,x 0 | 2 dx-t 2 * 2 * ∂Ω p(x)Q(x)|W ε,x 0 | 2 * ds x is increasing on the interval [0, X ε ] we have, by (2.38), M ε ≤ 1 2 X 2 ε Ω p(x)|∇W ε,x 0 | 2 dx - X 2 * ε 2 * ∂Ω p(x)Q(x)|W ε,x 0 | 2 * ds x - Ω F (x, t ε W ε,x 0 )dx.
Using (2.12)-(2.15) and the fact that

Ω |W ε,x 0 | 2 dx = o(ε), we obtain (2.39) M ε ≤ 1 2(N -1) p(x 0 ) (Q(x 0 )) N-2 S N -1 1 -Ω G(x, t ε W ε,x 0 )dx -      H(x 0 ) A 1 Q(x 0 )B 1 2 2 * -2 A ′ 2 ε| log ε| + o(ε| log ε|) if N = 3, H(x 0 ) p(x 0 ) 2 A 2 -2 2 * A 1 B 2 B 1 ε + o(ε) if N ≥ 4.
At this stage, we distinguish two cases:

When H(x 0 ) ≤ 0.
ε N-2 2 +∞ ε 1 2 t N -1 +∞ 0 G t -(N -2) (1 + r 2 ) (N-2) 2 r N -2 drdt = +∞.
and for N = 3

(2.42) lim ε→0 ε 1 2 | ln(ε)| +∞ ε 1 2 t 2 +∞ 0 G t -1 (1 + r 2 ) 1 2 rdrdt = +∞.
Then condition (2.8) holds.

Proof.

From (2.40) and (2.11), for ε > 0 sufficiently small, we have

(2.43) Ω G(x, t ε W ε,x 0 )dx ≥ Ω G Aε N-2 2 (ε + x N ) 2 + |x ′ -x ′ 0 | 2 N-2 2 dx
for some constant A > 0. Inserting (2.43) into (2.39) we write (2.44)

M ε ≤ 1 2(N -1) p(x 0 ) (Q(x 0 )) N -2 S N -1 1 - Ω G Aε N-2 2 (ε + (x N -x 0N )) 2 + |x ′ -x ′ 0 | 2 N-2 2 dx -      H(x 0 ) A 1 Q(x 0 )B 1 2 2 * -2 A ′ 2 ε| log ε| + o(ε| log ε|) if N = 3, H(x 0 ) p(x 0 ) 2 A 2 -2 2 * A 1 B 2 B 1 ε + o(ε) if N ≥ 4.
Finally, we claim that

(2.45) lim ε→0 1 ε Ω G Aε N-2 2 (ε + x N ) 2 + |x ′ -x ′ 0 | 2 N-2 2 dx = +∞ If N ≥ 4,

and

(2.46) lim

ε→0 1 ε| log ε| Ω G Aε 1 2 (ε + x N ) 2 + |x ′ -x ′ 0 | 2 N-2 2 dx = +∞ If N = 3.
which implies, together with (2.44), that

M ε < 1 2(N -1) p(x 0 ) (Q(x 0 )) N-2 S N -1 1 for ε > 0 suffi- ciently small.
Verification of (2.45) and (2.46):

Ω G Aε N-2 2 (ε + x N ) 2 + |x ′ -x ′ 0 | 2 N-2 2 dx = ε N IR N + G Aε -N-2 2 (1 + y N ) 2 + |y ′ | 2 N-2 2 dy + O(1) = ε N ω +∞ 0 +∞ 0 (1 + y N ) N -1 G 1 (1 + y N ) N -2 Aε -N-2 2 [1 + r 2 ] N-2 2 r N -2 drdy N
where ω is the area of sphere S N -2 . Using the change of variable t = ε

1 2 (1 + y N ) we get Ω G Aε N-2 2 (ε + x N ) 2 + |x ′ -x ′ 0 | 2 N-2 2 dx = ε N 2 ω +∞ ε 1 2 +∞ 0 t N -1 G 1 t N -2 A (1 + r 2 ) N-2 2 r N -2 drdt.
Then

ε N-2 2 +∞ ε 1 2 t N -1 +∞ 0 G t -(N -2) (1 + r 2 ) (N-2) 2
r N -2 drdt = 0.

and for N = 3

(2.49) lim

ε→0 ε 1 2 | ln(ε)| +∞ ε 1 2 t 2 +∞ 0 G t -1 (1 + r 2 ) 1 2 rdrdt = 0.
Then condition (2.8) holds.

Proof.

The proof of this Lemma is similar to proof of Lemma 2.2.

2
Now let us give some examples for the nonlinear perturbation.

Examples of f :

If H(x 0 ) > 0 then the two functions g below satisfy the hypothesis of Lemma 2.2.

1) g(x, u) = g(u) = µ |u| r-2 u with µ > 0 and 2 * < r < 2 * .

2)

g(x, u) = g(u) =    (3 + γ)u 2+γ ln(u) + |u| 2+γ if u > 1 (3 + γ)|u| 2+γ | ln(u)| + |u| 2+γ if u < 1 with 0 < γ < 2 N -2 .
If H(x 0 ) < 0 then the two functions g below satisfy Lemma 2.3.

1) g(x, u) = g(u) = µ |u| r-2 u with µ ∈ IR and 2 < r < 2 * . 2) g(x, u) = g(u) = ± 5 2 |u| 3 2 + |u| 7 2 (1 + 5|u| 2 ) 2 .
3 Existence results in presence of two critical exponents.

We assume that β = 1 and, as in the previous section, the nonlinearity f (x, u) satisfies the following basic assumptions. Moreover we assume that

(3.1) f (x, u) = a(x)u + g(x, u), with (3.2) a(x) ∈ L ∞ (Ω), ( 3
(3.5) λ 1 (a) = inf Ω |∇u| 2 -a(x)u 2 dx, Ω u 2 dx = 1 > 0. Set F (x, u) = u 0 f (x, t)dt for x ∈ Ω, u ∈ IR. Let define, for u ∈ H 1 (Ω), (3.6) Φ(u) = 1 2 Ω p(x)|∇u| 2 dx - 1 2 * Ω |u| 2 * dx - 1 2 * ∂Ω p(x)Q(x)|u| 2 * ds x - Ω F (x, u)dx.
Our main result in this section is and for some constant C (depending on ε). Therefore we have

(3.8) G(x, u) ≤ 1 2 a(x)u 2 + ε 2 u 2 + C 2 * |u| 2 *
for a.e x ∈ Ω, and for all u ∈ IR.

Therefore, for all u ∈ H 1 (Ω),

Φ(u) ≥ 1 2 Ω p(x)|∇u| 2 dx- 1 2 Ω a(x)|u| 2 dx- ε 2 Ω |u| 2 dx- C 2 * Ω |u| 2 * dx- 1 2 * ∂Ω p(x)Q(x)|u| 2 * ds x
Using (3.5) we easily see that, for ε > 0 small enough , there exist constants k > 0,

C 1 > 0 and C 2 > 0 such that Φ(u) ≥ k u 2 H 1 -C 1 u 2 * H 1 -C 2 u 2 * H 1 ≥ u 2 H 1 k -C 1 u 2 * -2 H 1 -C 2 u 2 * -2 H 1 for all u ∈ H 1 .
Which implies, since 2 * > 2 and 2 * > 2, that for some small α > 0 there exists ρ > 0 such that Φ(u) ≥ ρ, provided u = α.

On the other hand, for any u ∈ H 1 (Ω), u ≡ 0 in Ω, we have by (3.4) lim t→+∞ Φ(tu) = -∞. Thus for later purpose we take v = t 0 U ε,x 0 , where t 0 > 0 is chosen large enough so that v ∈ U and Φ(v) ≤ 0. Set

(3.9) c = inf P∈A max w∈P Φ(w),
where A denotes the class of continuous paths joining 0 to v.

Looking at (3.7) we see that c < M (S, S 1 ). By a result of Ambrosetti and Rabinowtz [START_REF] Ambrosetti | Dual variational methods in critical point theory and applications[END_REF], see also [START_REF] Brezis | Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents[END_REF], there exists a sequence {u j } in

H 1 (Ω) satisfying (3.10) Φ(u j ) → c < M (S, S 1 ) and (3.11)
Φ ′ (u j ) → 0 in H -1 (Ω)
Using (3.1) and (3.5), from (3.10) and (3.11) we write

(3.12) 1 2 u j 2 - 1 2 * Ω |u j | 2 * dx - 1 2 * ∂Ω p(x)Q(x)|u j | 2 * ds x - Ω G(x, u j ) = c + o (1), and (3.13)
u j 2 - Ω |u j | 2 * dx - ∂Ω p(x)Q(x)|u j | 2 * ds x - Ω g(x, u j )u j dx =< ξ j , u j > with ξ j → 0 in H -1 .
We start by showing that {u j } is bounded in H 1 (Ω). Computing (3.12)-1 2 * (3.13), we obtain

(3.14) 1 2(N -1) u j 2 + N -2 2N (N -1) Ω |u j | 2 * dx- Ω G(x, u j ) - 1 2 * g(x, u j )u j dx = c+o(1)+ < ξ j , u j > .
On the other hand, from (3.4) we have for all ε > 0 there exists C > 0 such that We deduce from (3.14)-(3.16), after using the embedding L 2 (Ω) ֒→ L 1 (Ω) and

H 1 (Ω) ֒→ L 2 (Ω) that, for ε > 0 small enough, 1 2(N -1) u j 2 + N -2 2N (N -1) (1 + ε) Ω |u j | 2 * dx -C ′ u j ≤ c + o(1)
for some constant C ′ > 0. This gives that {u j } is bounded in H 1 (Ω), otherwise we obtain a contradiction.

Extract a subsequence, still denoted by u j , such that

u j ⇀ u weakly in H 1 (Ω), u j → u strongly in L t (Ω) for all t < 2 * = 2N N -2 , u j → u a.e. on Ω, f (x, u j ) ⇀ f (x, u) weakly in L 2 * 2 * -1 (Ω), u j ⇀ u weakly in L 2 * (∂Ω), u j ⇀ u weakly in L 2 * (Ω).
We shall now verify that u ≡ 0 on Ω. Indeed , suppose that u ≡ 0. We claim that (3.17)

Ω f (x, u j )u j dx → 0 and Ω F (x, u j )dx → 0.
From (3.15) and (3.16), we have, for all ε > 0 there exists C > 0 such that

Ω f (x, u j )u j dx ≤ ε Ω |u j | 2 * dx + C Ω |u j |dx and Ω F (x, u + j )dx ≤ ε 2 * Ω |u j | 2 * dx + C 2 Ω |u j | 2 dx.
Since {u j } remains bounded in L 2 * (Ω) and u j → 0 in L 2 (Ω) we obtain (3.17). Now, extruding a subsequence, still denoted by u j , we may assume that there exist some constants l ≥ 0, m 1 ≥ 0 and m 2 ≥ 0 such that (3.18)

Ω p(x)|∇u j | 2 dx → l, Ω |u j | 2 * dx → m 1 , and ∂Ω p(x)Q(x)|u j | 2 * ds x → m 2 .
Passing to the limit in (3.12) and (3.13), we get

(3.19) 1 2 l - 1 2 * m 1 - 1 2 * m 2 = c and l -m 1 -m 2 = 0.
From the result of [24, Theorem 01], we know that there exists a constant C(Ω) > 0 such that for every w ∈ H 1 (Ω)

Ω |∇w| 2 dx + C(Ω) Ω |w| k dx ≥ S 2 2 N Ω |w| 2 * dx 2 2 * , with k = 2N N -1 if N ≥ 4 and k > 3 = 2N N -1 if N = 3.
We apply this result for w j = (p(x)) 1 2 u j and in particular for N = 3 we take k such that 6 = 2N N -2 > k > 3, we obtain for j large enough

Ω |∇(p(x)) 1 2 u j | 2 dx + C(Ω) Ω |(p(x)) 1 2 u j | k dx ≥ S 2 2 N Ω |(p(x)) 1 2 u j | 2 * dx 2 2 *
Since k < 2N N -2 for every N ≥ 3, thanks to the compact embedding H 1 (Ω) ֒→ L k (Ω), we have u j → 0 strongly in L k (Ω) and we deduce

Ω p(x)|∇u j | 2 dx + o(1) ≥ S 2 2 N Ω |(p(x)) 1 2 u j | 2 * dx 2 2 * + o(1).
Using the fact that p(x) ≥ p 0 for all x ∈ Ω, we see that

Ω p(x)|∇u j | 2 dx + o(1) ≥ p 0 S 2 2 N Ω |u j | 2 * dx 2 2 *
+ o(1).

At the limit we obtain

(3.20) (m 1 ) 2 2 * p 0 S 2 2 N ≤ l.
On the other hand, by the same way, from [24, Theorem 02] we have (see (2.29) for more details)

(3.21) (m 2 ) 2 2 * p(x 0 ) (Q(x 0 )) N -2 1 N-1 S 1 ≤ l.
                               1 2(N -1) l + N -2 2N (N -1) m 1 = c 1 N l - N -2 2N (N -1) m 2 = c m 1 ≤ 2 2 N l p(a)S 2 * 2 m 2 ≤   l [ p(x 0 ) (Q(x 0 )) N-2 ] 1 N-1 S 1   2 * 2 .
An easy computation yields

(3.23) 1 N l - N -2 2(N -1)N ( l p(x 0 ) (Q(x 0 )) N-2 S 1 ) 2 * 2 ≤ c ≤ 1 2(N -1) l + N -2 2(N -1)N ( 2 2 N l p 0 S ) 2 * 2 .
We can write

l ≤ ( l p(x 0 ) (Q(x 0 )) N-2 S 1 ) 2 * 2 + ( 2 2 N l p 0 S ) 2 * 2 .
If l = 0 then, since c > 0, we obtain a contradiction and we get the desired result. Now, if l = 0 we reduce to the study of the following polynomial

1 (2 -2 N p 0 S) N N-2 t 2 + 1 ( p(x 0 ) (Q(x 0 )) N-2 S 1 ) N-1 N-2 t -1 ≥ 0 where t = l 1 N-2 . Which is possible if t ≥ 2( p(x 0 ) (Q(x 0 )) N-2 S 1 ) N-1 N-2 1 + √ 1 + 4E ′ where E ′ =   p(x 0 ) (Q(x 0 )) N-2 S N -1 1 (2 -2 N p 0 S) N 2   2 N-2 .
From the left inequality of (3.23) and the fact that l = t N -2 , we obtain c ≥ M (S, S 1 ) which gives a contradiction with (3.10). Consequently u ≡ 0 and u is a solution of (1.1).

Remark 3.1

If we assume that

(3.24) F (x, v) ≤ 1 2 f (x, v)v + 1 N |v| 2 *
, for all v ∈ IR and for for a.e x ∈ Ω.

then the previous sequence {u j } is relatively compact in H 1 (Ω).

Let {u j } be the sequence defined in the proof of Theorem 3.1, we recall that u j converge weakly to u in H 1 (Ω). We will show that u j converges strongly to u in H 1 (Ω). Firstly, since u is a solution of (1.1), we have

Ω p(x)|∇u| 2 dx = Ω |u| 2 * dx + Ω f (x, u)udx + ∂Ω p(x)Q(x)|u| 2 * ds x . Therefore Φ(u) = Ω 1 N |u| 2 * + 1 2 f (x, u)u -F (x, u) dx + 1 2(N -1) ∂Ω p(x)Q(x)|u| 2 * ds x .
Using (3.24) we have Φ(u) ≥ 0. Now, we set v j = u ju. We write

(3.25) Ω p(x)|∇u j | 2 dx = Ω p(x)|∇u| 2 dx + Ω p(x)|∇v j | 2 dx + o(1)
and from [START_REF] Brezis | A relation between pointwise convergence of functions and convergence of functionals[END_REF] we deduce that

(u) + 1 2 Ω p(x)|∇v j | 2 dx - 1 2 * Ω |v j | 2 * dx - 1 2 * ∂Ω p(x)Q(x)|v j | 2 * ds x = c + o(1)
and (looking at (3.11))

(3.29) Ω p(x)|∇v j | 2 dx - Ω |v j | 2 * dx - ∂Ω p(x)Q(x)|v j | 2 * ds x = o(1)
. Now, we assume (for a subsequence) that exists some constants l ≥ 0, m 1 ≥ 0 and m 2 ≥ 0 such that

Ω p(x)|∇v j | 2 dx → l, Ω |v j | 2 * dx → m 1 and ∂Ω p(x)Q(x)|v j | 2 * ds x → m 2 .
Passing to limit in (3.28) and (3.29), using the Sobolev embedding, a easy computation yields

                         1 2(N -1) l + N -2 2N (N -1) m 1 = c -Φ(u) 1 N l -N -2 2N (N -1) m 2 = c -Φ(u) m 1 ≤ l p(a)S 2 * 2 m 2 ≤ l [ p(x 0 ) (Q(x 0 )) N-2 ] 1 N-1 S 1 2 * 2 .
Therefore, as in end of proof of Theorem 3.1, if l = 0 then c -Φ(u) ≥ M (S, S 1 ) which is a contradiction since c < M (S, S 1 ) and Φ(u) ≥ 0. Consequently l = 0 and then u j → u strongly in H 1 (Ω).

3.1 Sufficient conditions on f (x, u) which give condition (3.7):

We recall

S = inf IR N |∇u| 2 dx; u ∈ H 1 (IR N ), IR N |u| 2 * dx = 1 .
We consider, for all ε > 0, the following functions

(3.30) U ε,y (x) = ε ε 2 + |x ′ -y ′ | 2 + |x N -y N + µ(N -2) -1 ε| 2 N-2 2
, where x = (x ′ , x N ), y = (y ′ , y N ) ∈ IR N -1 ×]0, +∞[, µ ∈ IR and u ε,x 0 = ξ(x)U ε,x 0 (x), where ξ be a radial C ∞ -function such that, for a fixed positive constant R,

ξ(x) = 1 if |x -x 0 | ≤ R 4 0 if |x -x 0 | > R 2
It is known, see [START_REF] Chipot | On the solutions to some elliptic equations with nonlinear Neumann boundary conditions[END_REF] and [START_REF] Li | Uniqueness theorems through the method of moving spheres[END_REF], that U ε,y is a solution of the following problem

(3.31)          -∆u = N (N -2)u N+2 N-2 in IR n + u > 0 in IR N + -∂u ∂x N = µ u N N-2 on ∂IR N + = IR N -1 .
We draw on estimates made in [11, pages 17-22], we write (3.32)

Ω p(x)|∇u ε,x 0 | 2 dx = p(x 0 )A µ -µH(x 0 )p(x 0 )    K 1 ε + o(ε) if N ≥ 4 K 0 ε| log ε| + o(ε| log ε|) If N = 3, (3.33) Ω |u ε,x 0 | 2 * dx = B µ -µH(x 0 )K 2 ε + o(ε) for all N ≥ 3, (3.34) ∂Ω p(x)Q(x)|u ε,x 0 | 2 * ds x = p(x 0 )Q(x 0 )C µ +µH(x 0 )p(x 0 )Q(x 0 )K 3 ε+o(ε) for all N ≥ 3, Hence h(t µ ) = t 2 µ p(x 0 )A µ N - N -2 2N (N -1) p(x 0 )Q(x 0 )C µ t 2 N-2 µ .
By a standard computation we have

h(t µ ) = A µ   2A 0 Q(x 0 )C 0 1 + 1 + 4 AµBµ p(x 0 )(Q(x 0 )) 2 C 2 µ   N -2   1 N - N -2 N (N -1) 1 1 + 1 + 4 AµBµ p(x 0 )(Q(x 0 )) 2 C 2 µ   From (3.35)-(3.37) we see, for µ > 0 small enough, that h(t µ ) = A 0   2A 0 Q(x 0 )C 0 1+ 1+4 A 0 B 0 p(x 0 )(Q(x 0 )) 2 C 2 0   N -2   1 N -N -2 N (N -1) 1 1+ 1+4 A 0 B 0 p(x 0 )(Q(x 0 )) 2 C 2 0   +µL + o(µ),
where L is a constant. Using the fact that

S 1 = A 0 (C 0 ) 2 2 * , S = A∞ (B∞) 2 2 * , A ∞ = 2A 0 and B ∞ = 2B 0 , we obtain, for µ > 0 small enough, that max t≥0 J(t U ε,0 ) = h(t µ ) = M (S, S 1 ) + µ L + o(µ).
This gives the desired result.

2

Now, we will show, under some additional conditions on f (x, u), that u ε,x 0 , defined by (3.30), satisfies condition (3.7). We have Since f (x, u) is a lower-order perturbation of |u| 2 * -1 , we see that lim t→+∞ Φ(tu ε,x 0 ) = -∞.

Φ(tu ε,x 0 ) = 1 2 t 2 Ω p (
Therefore sup t≥0 Φ(tu ε,x 0 ) is achieved at some t ε ≥ 0 and t ε is bounded in IR + .

From now we suppose that t ε > 0, otherwise condition (3.7) is easily satisfied. We write t ε = t 0 + O(ε) when N ≥ 4 and t ε = t 0 + O(ε| ln(ε)|) when N = 3, using (3.32)-(3.34) we get

If N ≥ 4: Φ(t ε u ε,x 0 ) = t 2 ε 2 p(x 0 ) IR N + |∇U ε,0 | 2 dx - t 2 * ε 2 IR N + |U ε,0 | 2 * dx - t 2 * ε 2 p(x 0 )Q(x 0 ) R N-1 |U ε,0 (x ′ , 0)| 2 * dx ′ - t 2 0 2 µH(x 0 )K 1 ε + t 2 *
R N + |∇U 1,0 | 2 dx - t 2 * -2 0 p(x 0 ) IR N + |U 1,0 | 2 * dx -Q(x 0 )t 2 * -2 0 R N-1 |U 1,0 | 2 * dx ′ = 0.
On the other hand, since U 1,0 is a solution of (3.31) we see that where G(x, s) = s 0 g(x, r)dr. Now, we are able to give sufficient conditions on f to have the condition (3.7):

  for a.e. x ∈ Ω, and for all u ∈ IR, and then |F (x, u)| ≤ C 1 s |u| s + C 2 |u| for a.e x ∈ Ω, and for all u ∈ IR.

. 3 )

 3 |g(x, u)| = o(|u|) as u → 0, uniformly in x, (3.4) |g(x, u)| = o(|u| 2 * -1 ) as |u| → +∞, uniformly in x.

Theorem 3. 1

 1 Assume (3.1)-(3.5) and suppose, moreover, that(3.7)there exists somev 0 ∈ H 1 , v 0 ≥ 0 on Ω, v 0 = 0 on ∂Ω, such that sup t≥0 Φ(tv 0 ) < M (S, S 1 ), where M (S, S 1 ) is defined in(1.4).Then, problem (1.1) possesses a solution.Proof of Theorem 3.1. From (3.3) we have, for any ε > 0, there is a δ > 0 such that |g(x, u)| ≤ ε|u| for a.e x ∈ Ω, and for all |u| ≤ δ, thus, by (3.4), we obtain |g(x, u)| ≤ ε|u| + C|u| 2 * -1 for a.e x ∈ Ω, and for all u ∈ IR,

(3. 15 )≤ ε 2 *

 152 |g(x, u)| ≤ ε|u| 2 * -1 + C for a.e x ∈ Ω and for all u ∈ IR, |u| 2 * + Cu for a.e x ∈ Ω and for all u ∈ IR.

Combining ( 3 .

 3 [START_REF] Pierotti | On a Neumann problem involving two critical exponents: remarks on geometrical and topological aspects[END_REF],(3.20) and (3.21) we obtain the following(3.22)

Ω

  |u j | 2 * dx = Ω |u| 2 * dx + Ω |v j | 2 * dx + o(1), and (3.27) ∂Ω p(x)Q(x)|u j | 2 * ds x = ∂Ω p(x)Q(x)|u| 2 * ds x + ∂Ω p(x)Q(x)|v j | 2 * ds x + o(1).Inserting (3.25), (3.26) and (3.27) into (3.12) and (3.13) we get (3.28) Φ

0 2 *-F 2 *- t 2 * 0 2 *R N- 1 |U 2 0 2 FF 2 *≥ 4 t 2 0 2 p

 2221222422 µH(x 0 )K 2 ε (x, tu ε,x 0 )dx + o(ε) ≤ max t≥0 J(tU ε,0 ) -t 2 0 2 µH(x 0 )p(x 0 )K 1 ε + t 2 * 0 µH(x 0 )K 2 ε µH(x 0 )p(x 0 )Q(x 0 )K 3 ε -Ω F (x, tu ε,x 0 )dx + o(ε), If N = 3: Φ(t ε u ε,x 0 ε,0 (x ′ , 0)| 2 * dx ′ -t µH(x 0 )K 0 ε| ln(ε)| -Ω F (x, tu ε,x 0 )dx + o(ε| ln(ε)|) 0 )p(x 0 )K 0 ε| ln(ε)| -Ω (x, tu ε,x 0 )dx + o(ε| ln(ε)|). Therefore (3.41) Φ(t ε u ε,x 0 ) ≤ M 1 (S, S 1 ) -Ω (x, tu ε,x 0 )dx + o(µ) -µH(x 0 ) p(x 0 )Q(x 0 )K 3 ]ε + o(ε) if N (x 0 )K 0 ε| ln(ε)| + o(ε| ln(ε)|) if N = 3. Now,we need to give a explicit form of t 0 . Since sup t≥0 Φ(tu ε,x 0 ) = sup t≥0 h(t) is achieved at t ε then h ′ (t ε ) = 0 and letting ε → 0 we get (3.42)

R N- 1 |U 1 , 0 | 2 * 1 2 2 0 2 p 4 NF≥ 4 K 0 4 K 0

 110212244040 0 | 2 dx -N (N -2)IR N + |U 1,0 | 2 * dxµ dx ′ = 0.Combining(3.42) and (3.43) we obtain t 0 = (p(x 0 ) N (N -2)) * -2 . Using (3.38) and (3.39), for N ≥ 4, we see thatt (x 0 )K 1p(x 0 )Q(x 0 )K 3 = p(x 0 ) (N -2) 2 2 -3 + 2(N -1) (N -3)(N -2) 2 µ 2 +2 (N -1) (N -2) N (N -2) p(x 0 )p(x 0 )Q(x 0 )µ = K > 0.Combining this with (3.41) we obtain (3.44)Φ(t ε u ε,x 0 ) ≤ M 1 (S, S 1 ) -Ω (x, tu ε,x 0 )dx -µH(x 0 ) ε| ln(ε)| + o(ε| ln(ε)|) if N = 3.Using (3.1) and the fact that Ω |u ε,x 0 | 2 dx = o(ε), (3.44) becomes (3.45) Φ(t ε u ε,x 0 ) ≤ M 1 (S, S 1 ) -Ω G(x, tu ε,x 0 )dx -µH(x 0 ) ε| ln(ε)| + o(ε| ln(ε)|) if N = 3.

  ).

	Inserting (2.30) and (2.31) into (2.19) and (2.20) we get
	(2.32)

  Assume

		u 0 g(t)dt satisfies, for N ≥ 4
	(2.48)	lim ε→0
	(2.45) and (2.46) are a consequence of (2.41) and (2.49).	2
	When H(x 0 ) > 0.
	Lemma 2.3

that f (x, u) satisfies (2.1)-

(2.5)

and (2.7) or (2.1)-(2.4) and (2.6)-(2.7). Suppose that there exists some continuous function g such that (2.47) g(x, u) ≥ g(u) for a.e. x ∈ Ω and for all u ∈ IR and the primitive G(u) =

  x)|∇u ε,x 0 | 2 dx -t 2 * 2 * Ω |u ε,x 0 | 2 * dx -t 2 * 2 * ∂Ω p(x)Q(x)|u ε,x 0 | 2 * ds x -

Ω

F (x, tu ε,x 0 )dx.

where A µ , B µ , C µ and K i > 0 for i ∈ {0, 1, 2, 3} are defined by (3.35)

with K 2 > 0 and K 0 > 0 are some constants. Let

We have the following result

, where M (S, S 1 ) is defined in (1.4).

Proof.

We have

Looking at the polynomial B µ l 2 + p(x 0 )Q(x 0 )C µ lp(x 0 )A µ we deduce that

.

Proposition 3.3

Assume that f (x, u) satisfies (3.1)- (3.5) and that H(x 0 ) > 0. Suppose that there exists some continuous function g(.) such that g(x, u) ≥ g(u) for a. e. x ∈ Ω and for all u ∈ IR and the primitive G(u) = u 0 g(t)dt satisfies :

Then condition (3.7) holds.

Proof.

The proof become directly from (3.45). 2

Example of f : All the assumptions of Proposition 3.3 are satisfied for the following functions:

1. g(x, u) = g(u) = ±|u| r-2 u with 2 < r < 2 * and u ∈ IR.

2. g(x, u) = g(u) = u 2 * -1 (2 * ln(u) -1) (ln(u)) 2 for u > 0.