On a nonlinear eigenvalue problem in Sobolev spaces with variable exponent

TL;DR

This paper investigates a class of nonlinear Dirichlet problems involving the p(x)-Laplace operator within Sobolev spaces with variable exponents, establishing the existence of weak solutions using variational methods.

Contribution

It extends the theory of Sobolev spaces with variable exponents to nonlinear eigenvalue problems and proves the existence of solutions via the Mountain Pass Theorem.

Findings

Existence of weak solutions for the p(x)-Laplace problem.

Application of Mountain Pass Theorem in variable exponent spaces.

Framework for analyzing nonlinear PDEs with variable growth conditions.

Abstract

We consider a class of nonlinear Dirichlet problems involving the $p(x)$--Laplace operator. Our framework is based on the theory of Sobolev spaces with variable exponent and we establish the existence of a weak solution in such a space. The proof relies on the Mountain Pass Theorem.