  Our main purpose in this paper is to investigate a supercritical Sobolev-type
inequality for the $k$-Hessian operator acting on
$\Phi^{k}_{0,\mathrm{rad}}(B)$, the space of radially symmetric $k$-admissible
functions on the unit ball $B\subset\mathbb{R}^{N}$. We also prove both the
existence of admissible extremal functions for the associated variational
problem and the solvability of a related $k$-Hessian equation with
supercritical growth.
