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

Abstract:

We consider non-negative distributional solutions $u\in C^1 (\overline {B_R } )$ to the equation $-\operatorname {div} [g(|\nabla u|)|\nabla u|^{-1} \nabla u ] = f(|x|,u)$ 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.