Extremal functions for a supercritical k-Hessian inequality of Sobolev-type

Abstract

Our main purpose in this paper is to investigate a supercritical Sobolev-type inequality for the $k$-Hessian operator acting on ${\Phi }_{0,\mathrm{rad}}^k(B)$, 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.