1          Introduction

Let Ω be a bounded open subset of IR^N , N ≥ 2, let ∂Ω

its boundary and p(x) ∈ C(Ω) with p(x) > 1. Let ν be a weight function in Ω, ie: ν measurable and strictly positive a.e. in Ω. We suppose furthermore, that the weight function satisfies also the integrability conditions defined in section 2.

Let us consider the following degenerate p(x)-elliptic

where A is a Leray-Lions operator defined from

     1,p(x)                                                                                 ^−1,p0(x)(Ω,ν^∗) , with ν^∗ =

W₀                    (Ω,ν) to its dual W

0

ν1−p (x), by :

Au = −diva(x,u,∇u).

where a is a Caratheodory function from´ Ω × IR × IR^N −→ IR^N who satisfies assumptions of growth, ellipticity and strict monotonicity, while the nonlinear term g: Ω×IR×IR^N −→ IR checks assumptions of sign and growth. We suppose moreover that g checks the following condition of coercivity:

⁽ ∃ ρ₁ > 0,∃ ρ₂ > 0 such that : for |s| ≥ ρ1, |g(x,s,ξ)| ≥ ρ2ν(x)|ξ|p(x)

We suppose also that the second member f belongs to L¹(Ω).

Let consider the following degenerate p(x)-elliptic

(1)

                            u,                        g(x,u,∇u) ∈ L1(Ω).

In the case where p is constant and without weight, there is a wide literature in which one can find existence results for problem (1). When the second member f belongs to W ^−1,p0 (Ω), A. Bensoussan, L. Boccardo and F. Murat [6] studied the problem and give an existence result. While if f ∈ L¹(Ω) the initiated basic works were given by H. Brezis and Strauss [9], L. Boccardo and T. Galloue¨t [7] also proved an existence result for (1), which was extended to the a unilateral case studied by A. Benkirane and A. Elmahi [5]. When g is not necessarily the null function, T. Del Vecchio [10] proved first existence result for problem (1) in the case where g does not depend on the gradient and then in V. M Monetti and L. Randazzo [16] using, in both works, the rearrangement techniques. Whoever in [1], Y. Akdim, E. Azroul and A. Benkirane treated the problem (1) within the framework

1,p of Sobolev spaces with weight W₀ (Ω,ω), but while keeping p constant.

1. Azroul, A. Barbara and H. Hjiej [2] studied (1) , in the nonclassical case by considering nonstandard

1,p(x)

Sobolev spaces without weight W₀ (Ω). See as well [3] where existence and regularity of entropy solutions was obtained for equation (1) with degenerated

second member.

Our objectif, in this paper, is to study equation (1) by adopting Sobolev spaces with weight ν(x), and

1,p(x) to variable exponents p(x), W₀    (Ω,ν). We prove that the problem (1) admits at least a solution u ∈

1,p(x)

W₀               (Ω,ν).

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