Dirichlet boundary value problem related to the $p(x)-$Laplacian with discontinuous nonlinearity

TL;DR

This paper establishes the existence of weak solutions for a Dirichlet boundary value problem involving the variable exponent p(x)-Laplacian with discontinuous nonlinearities, using degree theory and Hammerstein equations.

Contribution

It introduces a novel approach to handle discontinuous nonlinearities in p(x)-Laplacian problems via degree theory and Hammerstein equations.

Findings

Proved existence of weak solutions under non-standard growth conditions.

Handled discontinuous nonlinearities in the boundary value problem.

Applied degree theory to variable exponent problems.

Abstract

In this paper, we prove the existence of a weak solution for the Dirichlet boundary value problem related to the $p(x)-$Laplacian $$ -\mbox{div}(|\nabla u|^{p(x)-2}\nabla u)+u\in -[\underline{g}(x,u),\overline{g}(x,u)], $$ by using the degree theory after turning the problem into a Hammerstein equation. The right hand side $g$ is a possibly discontinuous function in the second variable satisfying some non-standard growth conditions..