Symmetry and stability of non-negative solutions to degenerate elliptic equations in a ball

TL;DR

This paper investigates the symmetry and stability of non-negative solutions to a class of degenerate elliptic equations in a ball, establishing conditions for radial symmetry and analyzing minimizers of associated variational problems.

Contribution

The paper proves radial symmetry of solutions under growth conditions and shows that minimizers of the related variational problem are radial, extending understanding of degenerate elliptic equations.

Findings

Solutions are radially symmetric under certain growth conditions.

Global and local minimizers of the variational problem are radial.

Solutions satisfy a local symmetry property.

Abstract

We consider non-negative distributional solutions $u\in C^1 (\bar{B_R } )$ to the equation $-\mbox{div} [g(|\nabla u|)|\nabla u|^{-1} \nabla u ] = f(|x|,u)$ in a ball $B_R$, with $u=0$ on $\partial B_R $, where $f$ is continuous and non-increasing in the first variable and $g\in C^1 (0,+\infty )\cap C[0, +\infty )$, with $g(0)=0$ and $g'(t)>0$ for $t>0$. According to a result of the first author, the solutions satisfy a certain 'local' type of symmetry. Using this, we first prove that the solutions are radially symmetric provided that $f$ satisfies appropriate growth conditions near its zeros. In a second part we study the autonomous case, $f=f(u)$. The solutions of the equation are critical points for an associated variation problem. We show under rather mild conditions that global and local minimizers of the variational problem are radial.