On a PDE involving the ${\cal A}_{p(\cdot)}$-Laplace operator

TL;DR

This paper proves the existence of solutions for a PDE involving a generalized variable exponent Laplace operator, using fixed point and variational methods within variable exponent function spaces.

Contribution

It introduces an existence proof for PDEs with a generalized ${ m div}(| abla u|^{p( abla u)-2} abla u)$ operator, extending previous results to more general variable exponent settings.

Findings

Existence of solutions established for the PDE with ${ m div}(| abla u|^{p( abla u)-2} abla u)$ operator.

Utilizes Schauder's fixed point theorem and variational methods.

Operates within variable exponent Lebesgue and Sobolev spaces.

Abstract

This paper establishes existence of solutions for a partial differential equation in which a differential operator involving variable exponent growth conditions is present. This operator represents a generalization of the $p(\cdot)$-Laplace operator, i.e. $\Delta_{p(\cdot)}u={\rm div}(|\nabla u|^{p(\cdot)-2}\nabla u)$, where $p(\cdot)$ is a continuous function. The proof of the main result is based on Schauder's fixed point theorem combined with adequate variational arguments. The function space setting used here makes appeal to the variable exponent Lebesgue and Sobolev spaces.