Entropy Solutions for Nonlinear Elliptic Anisotropic Homogeneous Neumann Problem

We prove the existence and uniqueness of entropy solution for nonlinear anisotropic elliptic equations with Neumann homogeneous boundary value condition for -data. We prove first, by using minimization techniques, the existence and uniqueness of weak solution when the data is bounded, and by approximation methods, we prove a result of existence and uniqueness of entropy solution. 1. Introduction Let be an open bounded domain of with smooth boundary. Our aim is to prove the existence and uniqueness of entropy solution for the anisotropic nonlinear elliptic problem of the form where the right-hand side and , are the components of the outer normal unit vector. For the rest of the functions involved in (1), we are going to enumerate their properties after we make some notations. For any , we set and we denote For the exponents, , with for every and for all , we put and . Now, we can give the properties of the rest of the functions involved in (1). We assume that for , the function is Carathéodory and satisfies the following conditions: is the continuous derivative with respect to of the mapping , , that is, such that the following equality and inequalities holds for almost every . There exists a positive constant such that for almost every and for every , where is a nonnegative function in , with . There exists a positive constant such that for almost every and for every , with and for almost every and for every . We also assume that the variable exponents are continuous functions for all such that where . We introduce the numbers A prototype example, that is, covered by our assumptions is the following anisotropic equation: Set , where . Then, we get the following equation. Actually, one of the topics from the field of PDEs that continuously gained interest is the one concerning the Sobolev space with variable exponents, or depending on the boundary condition (see [1–23]). In that context, problems involving the -Laplace operator or the more general operator were intensively studied (see [13]). At the same time, some authors was interested by PDEs involving anisotropic Sobolev spaces with variable exponent when the boundary condition is the homogeneous Dirichlet boundary condition (see [15, 16, 18, 20, 24–26]). In that context, the authors have considered the anisotropic -Laplace operator or the more general variable exponent anisotropic operator When the homogeneous Dirichlet boundary condition is replaced by the Neumann boundary condition, one has to work with the anisotropic variable exponent Sobolev space instead of . The main difficulty which appears