On critical exponents of a $k$-Hessian equation in the whole space

Abstract

In this paper, we study negative classical solutions and stable solutions of the following $k$-Hessian equation $$F_k(D^{2}V)=(-V{)}^{p}in{R}^n$$ with radial structure, where $n\ge 3$, $1<k<n/2$ and $p>1$. This equation is related to the extremal functions of the Hessian Sobolev inequality on the whole space. Several critical exponents including the Serrin type, the Sobolev type, and the Joseph-Lundgren type, play key roles in studying existence and decay rates. We believe that these critical exponents still come into play to research $k$-Hessian equations without radial structure.

18 pages