Symmetry results for the $p(x)$-Laplacian equation

TL;DR

This paper investigates symmetry properties of solutions to the nonlinear $p(x)$-Laplacian equation, proving the existence of partially symmetric solutions in axially symmetric domains and full radial symmetry for certain stable solutions.

Contribution

It establishes new symmetry results for solutions of the $p(x)$-Laplacian, including existence of mountain-pass solutions with partial symmetry and conditions for full radial symmetry.

Findings

Existence of mountain-pass solutions with partial symmetry in axially symmetric domains.

Semi-stable or non-degenerate solutions in a ball are radially symmetric.

Conditions under which solutions exhibit symmetry properties.

Abstract

We consider the Dirichlet problem for the nonlinear $p(x)$-Laplacian equation. For axially symmetric domains we prove that, under suitable assumptions, there exist Mountain-pass solutions which exhibit partial symmetry. Furthermore, we show that Semi-stable or non-degenerate smooth solutions need to be radially symmetric in the ball.