  We establish a supercritical Trudinger-Moser type inequality for the
$k$-Hessian operator on the space of the $k$-admissible radially symmetric
functions $\Phi^{k}_{0,\mathrm{rad}}(B)$, where $B$ is the unit ball in
$\mathbb{R}^{N}$. We also prove the existence of extremal functions for this
new supercritical inequality.
