Existence of a nontrival solution for Dirichlet problem involving p(x)-Laplacian

TL;DR

This paper proves the existence of solutions for a nonlinear Dirichlet problem involving the p(x)-Laplacian with nonsmooth potential, using nonsmooth critical point theory and variational Sobolev spaces.

Contribution

It introduces new existence results for p(x)-Laplacian problems with nonsmooth potentials using advanced nonsmooth analysis techniques.

Findings

Existence of solutions established under specific conditions.

Application of nonsmooth critical point theory to variable exponent problems.

Extension of variational methods to nonsmooth, variable exponent settings.

Abstract

In this paper we study the nonlinear Dirichlet problem involving p(x)-Laplacian (hemivariational inequality) with nonsmooth potential. By using nonsmooth critical point theory for locally Lipschitz functionals due to Chang and the properties of variational Sobolev spaces, we establish conditions which ensure the existence of solution for our problem.