Existence of Solutions for an Elliptic p ( x ) − Kirchhoff-type Systems in Unbounded Domain

with p and q are real valued functions satisfying 1 < p (x) , q (x) < N (N ≥ 2) for every x ∈ R , and M1 and M2 are continuous and bounded functions. We confine ourselves to the case where M1 = M2 = M for simplicity. Notice that the results of this paper remain valid for M1 6= M2 by adding some hypothesis on M1 and M2. The real valued function F ∈ C 1 ( R × R ) satisfies some assumptions. The unknown real valued functions u and v stay in appropriate spaces. The operator ∆p(x)u = div ( |∇u| p(x)−2 ∇u ) designates the p(x)-Laplacian. The problem (1.1) discribes the stationary version presented by Kirchhoff [16]. More precisely, Kirchhoff proposed the following model


Introduction
The purpose of this paper is to investigate existence results for the following class of nonlocal elliptic system in with p and q are real valued functions satisfying 1 < p (x) , q (x) < N (N ≥ 2) for every x ∈ R N , and M 1 and M 2 are continuous and bounded functions.We confine ourselves to the case where M 1 = M 2 = M for simplicity.Notice that the results of this paper remain valid for M 1 = M 2 by adding some hypothesis on M 1 and M 2 .The real valued function F ∈ C 1 R N × R 2 satisfies some assumptions.The unknown real valued functions u and v stay in appropriate spaces.The operator ∆ p(x) u = div |∇u| p(x)−2 ∇u designates the p(x)-Laplacian.
The problem (1.1) discribes the stationary version presented by Kirchhoff [16].More precisely, Kirchhoff proposed the following model The parameters in equation (1.2) have the following meanings: E is the Young modulus of the material, ρ is the mass density, L is the length of the string, h is the area of cross-section, and P 0 is the initial tension.
The study of elliptic problems involving p(x)−Laplacian has interested in recent years, for the existence of solutions see [1], [9] and [12], and the eigenvalue involving p (x) −Laplacian problems see [10] and [11].
In our context, the author in [4], obtained the existence and multiplicity of solutions for the vector valued elliptic system where Ω is bounded domain in R N , with smooth boundary ∂Ω, p (x) , q (x) ∈ C + Ω with 1 < p − = min The author apply the direct variational approach and the theory of the variable exponent Sobolev spaces.
In [3], the authors show, using the Ekeland variational principle, the existence of solution for the problem

Preliminary results
In this section we recall some definitions and basic properties of the variable exponent Lebesgue-Sobolev spaces and introduce some notations used below.Let Existence of Solutions for an Elliptic...

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Denote by S R N the set of measurable real-valued functions defined on R N .We introduce for p ∈ C + R N , the space equipped with the so called Luxemburg norm This is a Banach space, called generalized Lebesgue-space.Define the variable exponent Sobolev space Moreover, we recall some previous results.

Moreover for any
Let us now define the modular corresponding to the norm |.| p(x) by Proposition 2.3.( [8], [15]) For all u ∈ L p(x) R N , we have In addition, we have In particular, if p (x) = p is a constant, then .., then the following statements are mutually equivalent: (1) lim Let p * (x) be the critical Sobolev exponent of p (x) defined by for p(x) < N +∞ for p(x) ≥ N , and let C 0,1 R N be the Lipschitz-continuous functions space.
( 8], [7]) If p (x) ∈ C 0,1 + R N , then there exists a positive constant c such that for all u ∈ W 1,p(x) R N .
2) If p is continuous on Ω and s is a measurable function on Ω, with p(x) ≤ s (x) < p * (x) , ∀x ∈ Ω, then the embedding Existence of Solutions for an Elliptic...

Existence of solutions
The solution of (1.1) belongs to the product space In what follows, W p(x),q(x) denote W p(x),q(x) R N .
Definition 3.1.We say that (u, v) ∈ W p(x),q(x) is a weak solution of (1.1) if for all (z, w) The Euler-Lagrange functional associated to problem (1.1) is defined as

Hypotheses
In this paper, we will use the following assumptions.
(H2) There exist positive functions a i , b i such that: (H3) There exist constants R > 0, θ > 1 and µ < 1 − 1 θ , and a positive function small, we have F x, t (H5) There exists m 0 > 0, µ; 0 < µ < 1 such that m 0 ≤ M (t) and The following existence theorem is based on an important compactness property of functionals.We first prove some lemmas.
Proof: we have as in [5] F We consider the fact that W p(x) ֒→ L s(x)p(x) R N , for s (x) > 1, there exists and |v| In the other hand such that i = + if u p(x) > 1, and i = − if u p(x) < 1, c is positive constant.So, for all (u, v) ∈ W p(x),q(x) , 1 < γ 1 , γ 2 , µ 1 , µ 2 < inf {p (x) , q (x)} with (u, v) p(x) = ρ large enough, ✷ Lemma 3.3.Assume that (H1) -(H5) holds.Then there exists (e 1 , e 2 ) ∈ W p(x),q(x) with (e 1 , e 2 ) > ρ such that I (e 1 , e 2 ) < 0 Proof: From (H5), we can obtain for t > t 0 M (t) ≤ M (t 0 ) , A.Djellit and S. Tas    This equation is as an extension of the classical d'Alembert's wave equation by considering the effects of changes in the length of the strings during the vibrations.