On a supercritical k-Hessian inequality of Trudinger-Moser type and extremal functions

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