Classification of the structures of stable radial solutions for semilinear elliptic equations in $\bf R^N$

TL;DR

This paper classifies the structure of stable radial solutions for semilinear elliptic equations in high-dimensional space, establishing criteria for their existence based on the behavior of the nonlinearity and linking stable singular solutions to overall solution structure.

Contribution

It provides a comprehensive classification of stable radial solutions for supercritical nonlinearities in ${f R^N}$, including criteria for existence and nonexistence based on asymptotic analysis.

Findings

Criteria for existence of stable solutions based on $f'(u)F(u)$ limits

Classification of solution structures with respect to stability

Relation between stable singular solutions and overall solution structure

Abstract

We study the stability of radial solutions of the semilinear elliptic equation $\Delta u +f(u)=0$ in ${\bf R^N}$, where $N \geq 3$ and $f$ is a general superciritical nonlinearity. We give a classification of the solution structures with respect to the stability of radial solutions, and establish criteria for the existence and nonexistence of stable radial solutions in terms of the limits of $f'(u)F(u)$ as $u \to 0$ or $\infty$, where $F(u) = \int^{\infty}_u 1/f(t)dt$. Furthermore, we show the relation between the existence of singular stable solutions and the solution structure.