Some results for semi-stable radial solutions of $k$-Hessian equations

TL;DR

This paper investigates semi-stable radial solutions of $k$-Hessian equations, providing pointwise estimates, existence conditions, and asymptotic behavior analysis based on spatial dimension and growth functions.

Contribution

It offers new pointwise estimates and necessary conditions for semi-stable solutions of $k$-Hessian equations, including unbounded and bounded cases, in Euclidean space.

Findings

Derived pointwise estimates for solutions

Established necessary conditions for existence

Analyzed asymptotic behavior at infinity

Abstract

We devote this paper to study semi-stable nonconstant radial solutions of $S_k(D^2u) = w(|x|)g(u)$ on the Euclidean space $R^n$. We establish pointwise estimates and necessary conditions for the existence of such solutions (not necessarily bounded) for this equation. For bounded solutions we estimate their asymptotic behavior at infinity. All the estimates are given in terms of the spatial dimension $n$, the values of $k$ and the behavior at infinity of the growth rate function of $w$.