Adil Abbassi,Chakir Allalou and Abderrazak Kassidi

LMACS Laboratory,Mathematics Department,Faculty of Sciences and Techniques,Sultan Moulay Slimane University Beni-Mellal,BP:523,Morocco.

Abstract.In this paper,we are concerned with a show the existence of a entropy solution to the obstacle problem associated with the equation of the type:

where Ω is a bounded open subset of RN,N≥2,A is an operator of Leray-Lions type acting from into its dualand L1 ?deta.The nonlinear term g:Ω×R×RN?→R satisfying only some growth condition.

Key words:Entropy solutions,Anisotropic elliptic equations,weighted anisotropic variable exponent Sobolev space.

1 Introduction

Consider Ω be a bounded open subset ofwith for all x in Ω,

In the particular case when pi=p for any i∈{1,...,N},Yazough,Azroul and Redwane(see[16])have proved the existence of entropy solutions to problem like(P).Then,Azroul,Benboubker and Ouaro[6]have obtained the above results via penalization methods.

The study of(P)is a new and interesting topic when the data is in L1.One result on this topic can be found in[5,8,11],where the discussion was conducted in the framework of weighted anisotropic Sobolev space with variable exponent(we refer to[1,2,11]for more details),the notion of a entropy solution was introduced by Benilan et.al[7,9]and P.-L.Lions[14]in their study of the Boltzmann equation.We mention some works in the direction of the anisotropic space such as[4,8].

The aim of this paper is to extend the results in[5]to the anisotropic obstacle nonlinear elliptic problem.We want to prove only existence results,the uniqueness problem being a rather delicate one,this kind of problems still attracting the interest of the researchers(see[10,11,15]for a survey).One of the motivations for studying(P)comes from applications to elasticity as the equations that models the shape of an elastic membrane which is pushed by an obstacle from one side affecting its shape.The layout of the paper is presented as follows:Section 2 contains a brief discussion of variable exponent Lebesgue with weighted and the weighted anisotropic variable exponent Sobolev space,in Section 3 we introduce some useful technical lemmas,we prove our main result in Section 4.

2 Preliminaries

In this section,we state some elementary properties of the weighted variable exponent Lebesgue-Sobolev spaces which will be used in the next sections.The basic properties of the variable exponent Lebesgue-Sobolev spaces.Let Ω be a bounded open subset of RN(N≥2),we assume that the variable exponent p(·):→[1,+∞[is log-H ?lder continuous on Ω,that is there is a real constant c>0 such that for all x,y∈,x/=y with|x?y|

3 Technical lemmas

4 Main results

In this section we formulate and prove the main result of the paper.

Now,we give a definition of entropy solutions for our unilateral elliptic problem(P).