Existence of Multiple Solutions for a Class of Nonhomogeneous Problems with Critical Growth

In this paper, we study the existence and multiplicity of solutions for (p 1 (x), p 2 (x))-equation with critical growth. The technical approach is mainly based on the variational method combined with the genus theory.


Introduction
In this article, we are concerned with the following problem −∆ p 1 (x) u − ∆ p 2 (x) u − a(x)|u| m(x)−2 u = λ|u| q(x)−2 u + f (x, u) in Ω u = 0 on ∂Ω, (P λ ) where Ω ⊂ R N (N ≥ 1) is a bounded smooth domain, λ is a positive parameter and f : Ω × R → R is a continuous function which satisfies some assumptions provided later.Moreover, p 1 , p 2 , q ∈ C(Ω) and m(x) = max(p 1 (x), p 2 (x)) for all x ∈ Ω, such that where m * (x) = Nm(x) N−m(x) for all x ∈ Ω and the set In recent years, the study of various mathematical problems with variable exponent growth condition has been received considerable attention in recent years.A more and more important number of surveys and books dealing with this type of problems and their corresponding functional spaces setting have been published (see [1-4, 12, 16-20]).We also have to mention Email: massarmed@hotmail.comM. Massar the books [13] and [21] as important references in this field.This great interest may be justified by their various physical applications.In fact, there are applications concerning elastic mechanics [25], electrorheological fluids [23,24], image restoration [9], dielectric breakdown, electrical resistivity and polycrystal plasticity [6,7] and continuum mechanics [5].
It is well known that although most of the materials can be accurately modeled with the help of the classical Lebesgue and Sobolev spaces L p and W 1,p , where p is a fixed constant, but there are some nonhomogeneous materials, for which this is not adequate, e.g. the rheological fluids mentioned above, which are characterized by their ability to drastically change their mechanical properties under the influence of an exterior electromagnetic field.Thus it is necessary for the exponent p to be variable, hence the need for spaces with variable exponents.This leads, on the one hand,to many interesting applications, and, on the other hand,to the study of much more mathematically complicated problems.
In [19], Mihȃilescu considered the problem where m(x) = max{p 1 (x), p 2 (x)} for any x ∈ Ω or m(x) < q(x) < Nm(x) N−m(x) for any x ∈ Ω.In the first case, using mountain pass theorem, he established the existence of infinity many solutions.In the second case, by simple variational arguments, he proved that the problem has a solution for λ large enough.The novelty of this paper lies in the fact we consider problem (P λ ), with growth q(x) which is critical in a set with positive measure.The difficulty in this case, is due to the lack of compactness of the imbedding W 1,m(x) 0 (Ω) → L m * (x) (Ω) and the Palais-Smale condition for the corresponding energy functional could not be checked directly.To deal with this difficulty, we use a version of the concentration compactness lemma due to Lions for variable exponents [8].
Here, we are interested in the existence and multiplicity of weak solutions under the following hypotheses on a(x) and f .
(a 1 ) a(x) ∈ L ∞ (Ω) and there exists α > 0 such that Example 1.1.In this example we just exhibit a function a(x) satisfying assumption (a 1 ).

Preliminaries
Here, we state some interesting properties of the variable exponent Lebesgue and Sobolev spaces that will be useful to discuss problem (P λ ).Every where below we consider Ω ⊂ R N to be a bounded domain with smooth boundary and p(x) ∈ C + (Ω), where Define the variable exponent Lebesgue space by This space endowed with the Luxemburg norm, dx ≤ 1 is a separable and reflexive Banach space.Denoting by L p (x) (Ω) the conjugate space of L p(x) (Ω) where 1 p(x) + 1 p (x) = 1; for any u ∈ L p(x) (Ω) and v ∈ L p (x) (Ω) we have the following Hölder type inequality Now, we introduce the modular of the Lebesgue-Sobolev space L p(x) (Ω) as the mapping ρ p(x) : L p(x) (Ω) → R, defined by In the following proposition, we give some relations between the Luxemburg norm and the modular.
(2) There is a constant C > 0 such that

Proof of main result
We will start by recalling an important abstract theorem involving genus theory, which will be used in the proof of Theorem 3.2.Theorem 4.1 ([22]).Let E be an infinite dimensional Banach space with E = V X, where V is finite dimensional and let I ∈ C 1 (E, R) be a even function with I(0) = 0 and satisfying (i) There are constants β, > 0 such that I(u) ≥ β for all u ∈ ∂B ∩ X; (ii) There is τ > 0 such that I satisfies the (PS) c condition, for 0 < c < τ; Then 0 < β ≤ c j ≤ c j+1 for j > k, and if j > k and c j < τ, we have that c j is the critical value of I.
In the sequel, we derive some results related to the above theorem and the Palais-Smale compactness condition.
Since we will rely on the critical point theory, we define the energy functional corresponding to problem (P λ ) as Clearly, I λ is C 1 functional and the critical points of it are weak solutions of problem (P λ ).
Thanks to the continuity of f , there is On the other hand, by the first part of ( f 1 ), for each ε > 0 there exists 0 By integrating this last inequality, we get Therefore Hence for ε sufficiently small, Using the fact that it follows that By the continuous embedding W 1,m(x) 0 (Ω) → L q(x) (Ω), there exists C 1 > 0 such that Consequently, by Proposition 2.1, for u = , with 0 < < 1, Since m + < q − , there exists β > 0 such that I λ (u) ≥ β for u = , where is chosen sufficiently small.Lemma 4.3.Assume that (a 1 ), ( f 1 ) and (mq 1 ) hold.Then I λ satisfies condition (iii) given in Theorem 4.1.
Using (4.12), it follows that If ν j > 0 for some j ∈ J , by Lemma 4.5, we get which is impossible, and so ν j = 0 for all j ∈ J .Hence This implies lim n→+∞ Ω |u n − u| q(x) dx = 0, thanks to Proposition 2.1, we deduce By standard arguments, we see that On the other hand, we have Recalling the following well known inequality in R N , Divide Ω in two parts as follows:  .
Since {u n } is bounded in W  (Ω).

Proof of Theorem 3.2
By choosing for each k ≥ 1, λ k sufficiently small, we construct a sequence (λ k ), with λ k > λ k+1 such that M k < λ Therefore, for λ ∈ (λ k+1 , λ k ], Thanks to Theorem 4.1, the levels I λ has at least k critical points.Now, if c λ j = c λ j+1 for some j = 1, . . ., k − 1, again Theorem 4.1 implies that K c λ j is an infinite set [22,Chap. 7] and hence in this case, problem (P λ ) has infinitely many solutions.Conclusion, problem (P λ ) has at least k pair solutions.
and γ(Y) ≤ n − j , where Σ = {Y ⊂ E\{0} : Y is closed in E and Y = −Y} and γ(Y) is the genus of Y ∈ Σ.For each j ∈ N, let c j = inf K∈Γ j max u∈K I(u).