Radial symmetry of solutions to diffusion equations with discontinuous nonlinearities

TL;DR

This paper establishes radial symmetry of bounded nonnegative solutions to certain diffusion equations with discontinuous nonlinearities, extending previous methods and results to broader cases including all dimensions for specific nonlinearities.

Contribution

It introduces a new approach extending P. L. Lions' method to prove radial symmetry for solutions with discontinuous nonlinearities in various dimensions.

Findings

Radial symmetry holds for solutions with discontinuous nonlinearities in a ball.

New results for the case p=2 in all dimensions.

Radial symmetry proven for p ≥ n with locally bounded nonlinearities.

Abstract

We prove a radial symmetry result for bounded nonnegative solutions to the $p$-Laplacian semilinear equation $-\Delta_p u=f(u)$ posed in a ball of $\mathbb R^n$ and involving discontinuous nonlinearities $f$. When $p=2$ we obtain a new result which holds in every dimension $n$ for certain positive discontinuous $f$. When $p\ge n$ we prove radial symmetry for every locally bounded nonnegative $f$. Our approach is an extension of a method of P. L. Lions for the case $p=n=2$. It leads to radial symmetry combining the isoperimetric inequality and the Pohozaev identity.