  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
$\mathbb{R}^n$, $D^2u$ is the Hessian matrix of $u,\,g$ is a positive $C^1$
nonlinearity and $S_k\left(D^2u\right)$ 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.
