A priori estimates of stable solutions of the general Hardy-Henon equation in the ball

TL;DR

This paper derives a priori bounds and sharp pointwise estimates for semi-stable radial solutions of the Hardy-Henon equation in the unit ball, revealing dimension-dependent behaviors and constructing examples of unbounded solutions.

Contribution

It provides the first comprehensive a priori estimates and sharp pointwise bounds for semi-stable solutions of the Hardy-Henon equation in the ball, extending understanding of their stability and unboundedness.

Findings

Solutions are bounded for dimensions 2 to 10+4α.

Sharp pointwise estimates are established for higher dimensions.

A large family of unbounded semi-stable solutions is constructed.

Abstract

This paper is devoted to the study of semi-stable radial solutions $u\in H^1(B_1)$ of $-\Delta u=\vert x\vert^\alpha f(u) \mbox{ in } B_1\setminus\lbrace0\rbrace$, where $f\in C^1(\mathbb{R})$ is a general nonlinearity, $\alpha>-2$ and $B_1$ is the unit ball of $\mathbb{R}^N$, $N>1$. We establish the boudness of such solutions for dimensions $2\leq N<10+4\alpha$ and sharp pointwise estimates in the case $N\geq10+4\alpha$. In addition, we provide, for this range of dimensions, a large family of semi-stable radially decreasing unbounded $H^1(B_1)$ solutions.