Radial symmetry of solutions to diffusion equations with discontinuous 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.