Existence and Regularity of Solution for Strongly Nonlinear p(x)-Elliptic Equation with Measure Data

Abstract

The first part of this paper is devoted to study the existence of solution for nonlinear p(x) elliptic problem A(u)=μ in Ω, u =0 on ∂Ω, with a right-hand side measure, where Ω is a bounded open set of $\mathbb{R}$^N, N ≥ 2 and A(u)=-div(a(x,u,∇u)) is a Leray-Lions operator defined from W^1,p(x)₀ (Ω) in to its dual W^-1,p'(x)(Ω). However the second part concerns the existence solution, of the following setting nonlinear elliptic problems A(u)+g(x,u,∇u)=μ in Ω, u=0 on ∂Ω. We will give some regularity results for these solutions.