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

TL;DR

This paper investigates the existence and properties of solutions to a $k$-Hessian equation in the entire space, identifying critical exponents that influence solution behavior, with implications for related inequalities and non-radial cases.

Contribution

It characterizes critical exponents for $k$-Hessian equations with radial symmetry, linking them to solution existence, decay, and stability, and suggests their relevance beyond radial symmetry.

Findings

Identification of key critical exponents (Serrin, Sobolev, Joseph-Lundgren)

Analysis of solution decay rates and stability

Connection to extremal functions of Hessian Sobolev inequalities

Abstract

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.