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 \(\Phi ^{k}_{0,\textrm{rad}}(B)\) ${\mathrm{\Phi }}_{0,\text{rad}}^k(B)$ , where B is the unit ball in \({\mathbb {R}}^{N}\) ${\mathbb{R}}^N$ . We also prove the existence of extremal functions for this new supercritical inequality.