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

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