Radial symmetry of solutions to diffusion equations with discontinuous nonlinearities

Details Radial symmetry of solutions to diffusion equations with discontinuous nonlinearities Radial symmetry of solutions to diffusion equations with discontinuous nonlinearities

2011-01-26

arxiv.org/abs/1101.5094v1

Mathematics

Analysis of PDEs

Scientific paper

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.

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