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

Moulay Cherif Hassib; Youssef Akdim; Elhoussine Azroul; Abdelkrim Barbara

doi:10.4208/jpde.v30.n1.3

Authors

Moulay Cherif Hassib Laboratory LSI, Faculty Polydisciplinary of Taza. University Sidi Mohamed Ben Abdellah, P. O. Box 1223 Taza Gare, Marocco
Youssef Akdim Laboratory LSI, Faculty Polydisciplinary of Taza. University Sidi Mohamed Ben Abdellah, P. O. Box 1223 Taza Gare, Marocco
Elhoussine Azroul Laboratory, LAMA, Faculty of Science dhar El Mahraz Fez. University Sidi Mohamed Ben Abdellah, P. O. Box 1796 Atlas Fez, Morocco
Abdelkrim Barbara Laboratory, LAMA, Faculty of Science dhar El Mahraz Fez. University Sidi Mohamed Ben Abdellah, P. O. Box 1796 Atlas Fez, Morocco

DOI:

https://doi.org/10.4208/jpde.v30.n1.3

Keywords:

Sobolev spaces with variable exponents;strongly nonlinear p(x)-elliptic equations with measure data;regularity

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.