  In this paper, we study negative classical solutions and stable solutions of
the following $k$-Hessian equation $$ F_k(D^2V)=(-V)^p \quad in~R^n $$ with
radial structure, where $n \geq 3$, $1<k<n/2$ 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.
