Existence of Solutions For a Class of Strongly Coupled p(x)-laplacian System

abstract: The present work is concerned with the study of a strongly coupled nonlinear elliptic system on the whole space R involving the p(x)-laplacien operator. We employ variational methods and the theory of the variable exponent Sobolev spaces, in order to establish some sufficient conditions for the existence of non-trivial solutions.


Introduction
In this paper, we study the existence of nontrivial weak solutions for the following (p, q)-gradient elliptic system: Here p, q : Ω → R two functions of class C(Ω) such that 1 < p(x), q(x) < N (N ≥ 2) for all x ∈ R N and the coefficients a, b, are variables.The real-valued functions f, g are given functions and ∆ p(x) u is the p(x)-Laplacian operator defined by ∆ p(x) u := div(|∇u| p(x)−2 ∇u).The operator ∆ p(x) u := div(|∇u| p(x)−2 ∇u) is called p(x)−Laplace where p is a continuous non-constant function.This differential operator is a natural generalization of the p-Laplace operator ∆ p u := div(|∇u| p−2 ∇u), where p > 1 is a real constant.However, the p(x)−Laplace operator possesses more complicated nonlinearity than p-Laplace operator, due to the fact that ∆ p(x) is not homogeneous.This fact implies some diffculties; for example, we can not use the Lagrange Multiplier Theorem in many problems involving this operator.
The study of differential and partial differential involving variable exponent conditions is a new and an interesting topic.The interest in studying such problems was stimulated by their applications in elastic mechanics, fluid dynamics, electrorheological fluids, image processing, flow in porous media, calculus of variations, nonlinear elasticity theory, heterogeneous porous media models (see Acerbi-Mingione [1] , Diening [5], Ružicka [15], Zhikov [17]) etc....These physical problems were facilitated by the development of Lebesgue and Sobolev spaces with variable exponent.
In literature, elliptic systems with standard and nonstandard growth conditions have been studied by many authors.Let us briefly recall the literature concerning related elliptic systems.In [3,2] the authors show the existence of nontrivial solutions for the following p−Laplacian problem: where the p-Laplace operator ∆ p u := div(|∇u| p−2 ∇u), with p > 1, α, β > 0, p, q > 1, and is a real f, g are given functions.In [3], the author's obtain necessary and sufficient conditions on the coefficients for having a maximum principle for system (1.2).Then using the method of sup and super solutions, they prove the existence of positive solutions under some conditions on the functions f and g.In [2], the authors apply the theory of monotone operators to obtain the nontrivial solutions of the system (1.2).
In Khafagy-Serag [10] deal with the following problem: where the degenerate p-Laplacian defined as ∆ p,P u = div[P (x)|∇u| p−2 ∇u].Using an approximation method, they apply the Schauder's Fixed Point Theorem to get the nontrivial solutions of the system.Moreover, they gives necessary and sufficient strongly p(x)-Laplace system 185 conditions for having the maximum principle for this system.
In Djellit-Youbi-Tas [6] show the existence of nontrivial solutions for the following p(x)-Laplacian system: Here p, q : Ω → R two functions of class C(Ω) such that 1 < p(x), q(x) < N (N ≥ 2) for all x ∈ R N .However, the function Introducing some natural growth hypotheses on the right-hand side of the system which will ensure the semi-continuous and coercivity for the corresponding Euler-Lagrange functional of the system, the authors use critical point theory to obtain the existence of nontrivial weak solution of the system (1.3).In Ogras-Mashiyev-Avci-Yucedag [13] using a weak version of the Palais-Smale condition, that is, Cerami condition, they apply the mountain pass theorem to get the nontrivial solutions of the system (1.3).
In Xu-An [16] study the following elliptic systems of gradient type with nonstandard growth conditions The potential function F needs to satisfy Caratheodory conditions.Using critical point theory, they establish existence and multiplicity of solutions in sub-linear and super-linear cases.
Inspired by the above-mentioned papers, we deal with the existence of nontrivial solutions for system (1.1).We know that in the study of p(x)-Laplace equations in R N , a main difficulty arises from the lack of compactness.In this paper we will overcome this difficulty by establishing some growth conditions and regularity on the nonlinearities f and g, which will ensure the mountain pass geometry and Cerami condition for the corresponding Euler-Lagrange functional.By the mountain pass theorem, the basic results on the existence of solutions of system (1.1) will be presented.
The outline of this paper is as follows.In section 2, we will recall some basic facts about the variable exponent Lebesgue and Sobolev spaces which we will use later.Our main results are stated in Section 3. Proofs of our results will be presented in section 4.

Preliminary Results
To deal with the p(x)-Laplacian problem, we need introduce some functional spaces (Ω) and properties of the p(x)-Laplacian which we will use later.Denote by S(Ω) be the set of all measurable real-valued functions defined in Ω.Note that two measurable functions are considered as the same element of S(Ω) when they are equal almost everywhere.Set with the norm ) becomes a Banach space.We call it variable exponent Lebesgue space.Moreover, this space is a separable, reflexive and uniform convex Banach space; see [9, Theorems 1.6, 1.10, 1.14].
strongly p(x)-Laplace system 187 An important role in manipulating the generalized Lebesgue spaces is played by the modular of the L p(x) (Ω) space, which is the mapping ρ p(x) : L p(x) (Ω) → R defined by If (u n ), u ∈ L p(x) (Ω) and p + < ∞.Then the following relations hold true.
The following result generalizes the well-known Sobolev embedding theorem.

Main Results
Before stating our main results, we make the following assumptions throughout this paper: Moreover, there exists a function , Now we denote by E the product space D 1,p(x) × D 1,q(x) , defined as the completion of C ∞ 0 (R N ) with respect to the norm We remark that condition (B1) Then, for all w ∈ E, the following relations hold The main result of this paper is given by the following theorem: We point out the fact that the result of Theorem 3.1 extends the results from [12] , [14] where similar equations are studied in the case of p−laplacian operator.The energy functional corresponding to problem (1.1) is defined as Similar arguments as those used in [7] assure that I ∈ C 1 (E, R) with for all (Φ, Ψ) ∈ E. Thus, we observe that any critical points of the functional I are a weak solutions for problem (1.1).