A characterization of semistable radial solutions of k-Hessian equations

Abstract

We characterize semistable radial solutions of the equation $S_k\left(D^2u\right)=g(u)\;\mbox{in } B_1$, where $B_1$ is the unit ball of ${\mathbb{R}}^n$, $D^{2}u$ is the Hessian matrix of $u,g$ is a positive $C^1$ nonlinearity and $S_k\left(D^{2}u\right)$ denotes the $k$-Hessian operator of $u$. This class of radial solutions has been recently introduced by the authors in [8]. The proofs are new relative to those given in [8] and focus on the structure of the equation directly, thereby improving some previous results.