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

TL;DR

This work analyzes the stability of radial solutions to $\Delta u + f(u)=0$ in $\mathbb{R}^N$ with $N\ge 3$ for general supercritical nonlinearities $f$. It develops a stability-based classification of radial solutions, hinging on the limits $q_0=\lim_{u\to 0} f'(u)F(u)$ and $q_{\infty}=\lim_{u\to \infty} f'(u)F(u)$ relative to the Julia–Lindelöf threshold $q_{JL}$, and introduces generalized scaling to relate $f$- and $g$-nonlinearities. The paper establishes a complete trichotomy of stable-radial structures for $N\ge 11$, provides sharp existence and nonexistence criteria for stable radial solutions, and connects stability to the existence of singular radial solutions, with explicit nonlinear examples. This framework yields Liouville-type results, extremal-solution analogies, and practical criteria for constructing or ruling out stable radial states in high dimensions.

Abstract

We study the stability of radial solutions of the semilinear elliptic equation $Δu +f(u)=0$ 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.

Theorems & Definitions (63)

• Theorem 1.1
• Remark
• Theorem 1.2
• Remark
• Corollary 1.3
• Corollary 1.4
• Theorem 1.5
• Remark
• Theorem 1.6
• Corollary 1.7