On some multiple solutions for a $p(x)$-Laplace equation with supercritical growth

TL;DR

This paper investigates multiple solutions for a variable exponent p(x)-Laplacian problem with supercritical growth, extending existing results to include supercritical cases using advanced mathematical techniques.

Contribution

It introduces a novel approach combining truncation and De Giorgi iteration to establish the existence of at least three solutions in supercritical growth scenarios.

Findings

Established at least three solutions for the supercritical p(x)-Laplacian problem.

Extended previous subcritical and critical growth results to supercritical cases.

Applied Ricceri's principle with new techniques for multiplicity results.

Abstract

We consider the multiplicity of solutions for the $p(x)$-Laplacian problems involving the supercritical Sobolev growth via Ricceri's principle. By means of the truncation combining with De Giorgi iteration, we can extend the result about subcritical and critical growth to the supercritical growth and obtain at least three solutions for the $p(x)$ Laplacian problem.