Multiplicity of Solutions for Anisotropic Dirichlet Problem With Variable Exponent

: We establish some results on the existence of multiple nontrivial solutions for a general anisotropic elliptic equations. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with adequate variational methods and a variant of the Mountain Pass lemma.


Introduction
In the last few decades, one of the topics from the field of partial differential equations that has continuously attracted interest is that concerning the Sobolev space with variable exponents, W 1,p(.) (where p(.) is a function depending on x), see for example the monograph [4] and the references therein.Naturally, problems involving the p(x)-Laplacian operator were intensively studied.
On the other hand, it has been experimentally shown that the above-mentioned fluids may have their viscosity undergoing a significant change, see [1].Consequently, the mathematical modelling of such fluids requires the introduction of the so-called anisotropic variable spaces.Indeed, there is by now a large number of papers and increasing interest about anisotropic problems.With no hope of being complete, let us mention some pioneering works on anisotropic Sobolev spaces [12,15].Therefore, in the recent years, the study of various mathematical problems modeled by quasilinear elliptic and parabolic equations with both anisotropic and variable exponent has received considerable attention.
Let Ω ⊂ R N (N ≥ 2) be a bounded domain with smooth boundary.In this paper we study the following nonlinear anisotropic elliptic equations where △− → p (.) represents the − → p (.)-Laplace operator, that is, and for i = 1, ..., N , we assume that p i is a continuous function on Ω such that inf We set, For any h ∈ C + (Ω), we define Moreover, let's put the positive real numbers P + M , P + m , P − m which defined as the following Throughout this paper, we assume that The △− → p (.) -Laplacian problems on a bounded domain have been investigated and some interesting results have been obtained (see [6,8,13,14] and references therein).Inspired by the above references and the work in [7], we prove that there exist two nontrivial solutions for (P).
as t → 0 and uniformly for x ∈ Ω, with q − > P + M .Now, we can state the following result.
The △− → p (.) -laplacian operator possesses more complicated nonlinearities than the p -laplacian operator, mainly due to the fact that it is not homogeneous.This paper contains three sections.We will first introduce some basic preliminary results and lemmas in section 2. In section 3, we will give the proof of our main result.

Preliminary results
We recall in this section some definitions and basic properties of the variable exponent Lebesgue-Sobolev spaces L p(x) (Ω) and W 1,p(x) 0 (Ω), where Ω is a bounded domain in R N .Throughout this paper, we assume that p(x) > 1, p(x) ∈ C 0,α ( Ω) with α ∈ (0, 1).For any p(x) ∈ C + ( Ω), we define the variable exponent Lebesgue space We define a norm, the so-called Luxemburg norm, on this space by the formula Variable exponent Lebesgue space resemble classical Lebesgue space in many respects: they are Banach spaces, the Hölder inequality holds, they are reflexive if and only if 1 < p − ≤ p + < ∞ and continuous functions are dense, if p + < ∞.The inclusion between Lebesgue spaces also generalizes naturally: if 0 <| Ω |< ∞ and p 1 , p 2 are variable exponents so that p 1 (x) ≤ p 2 (x) almost everywhere in Ω then there exists the continuous embedding L p2(x) (Ω) ֒→ L p1(x) (Ω).We denote by L q(x) (Ω) the conjugate space of L p(x) (Ω), where 1 p(x) + 1 q(x) = 1.For any u ∈ L p(x) (Ω) and v ∈ L q(x) (Ω), the Hölder type inequality holds true.For more details, we can refer to [11].
An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the Modular of the L p(x) (Ω) space, which is the mapping J : L p(x) (Ω) → R defined by (Ω) and p + < ∞ then the following relations hold true Spaces with p + = ∞ have been studied by [5].
Recall that the weak solutions of (P) are the critical points of the associated energy functional Φ, given by It is well known that under (F 0 ), Φ is well defined and is a C 1 functional with derivative given by (Ω).Now, we consider the truncated problem where We denote by u + = max(u, 0) and u − = max(−u, 0) the positive and negative parts of u.
We need the following lemmas.
Similarly, nontrivial critical points of Φ − are non-positive solutions of (P).This ends the proof.

Proof of main result
To apply the mountain pass theorem, we will do separate studies of the compactness of Φ ± and its geometry.
Proof.Let (u n ) n be a (PS) sequence for the functional Φ (Ω).Using the hypothesis (F 1 ), since Φ + (u n ) is bounded, we have where C 1 and C 2 are two constants.Note that Suppose, by contradiction that (u n ) n unbounded in W 1, − → p (.) 0 (Ω), so u n ≥ 1 for rather large values of n.For each i ∈ {1, ..., N } and n we define (Ω).The proof of lemma 3.1 is complete.

Lemma 3.2.
There exist r > 0 and α > 0 such that Proof.The conditions (F 0 ) and (F 2 ) assure that For u small enough, we have For such an element u we have |∂ xi u| pi(.) < 1 and , by relation (2.2), we obtain By the condition (F 0 ), it follows (Ω).Since u is small enough, we deduce ) is strictly positive in a neighborhood of zero.
It follows that there exist r > 0 and α > 0 such that The proof is completed.Then, the functional Φ + has a critical point u + with Φ + (u + ) ≥ α.But, Φ + (0) = 0, that is, u + = 0. Therefore, the problem (P + ) has a nontrivial solution which, by lemma 2.2, is a non-negative solution of the problem (P).
Similarly, using Φ − , we show that there exists furthermore a non-positive solution.The proof of theorem 1.1 is now complete.
Multiplicity of Solutions for Anisotropic Dirichlet Problem With Variable Exponent 7 with a continuous and compact embedding, what implies the existence of C 4 , C 5 > 0 such that u