On weak solutions to a fractional Hardy-H\'enon equation: Part II: Existence

TL;DR

This paper investigates the existence of stable weak solutions to a fractional Hardy-Hénon equation, demonstrating the existence of positive radial solutions in critical and supercritical cases, and revealing multiple Joseph-Lundgren critical exponents.

Contribution

It establishes the existence of stable solutions for critical and supercritical exponents and identifies multiple Joseph-Lundgren critical exponents for certain parameters.

Findings

Existence of positive radial stable solutions in critical and supercritical regimes.

Multiple Joseph-Lundgren critical exponents for specific parameter ranges.

Non-existence of this property when s=1.

Abstract

This paper and [29] treat the existence and nonexistence of stable weak solutions to a fractional Hardy--H\'enon equation $(-\Delta)^s u = |x|^\ell |u|^{p-1} u$ in $\mathbb{R}^N$, where $0 < s < 1$, $\ell > -2s$, $p>1$, $N \geq 1$ and $N > 2s$. In this paper, when $p$ is critical or supercritical in the sense of the Joseph--Lundgren, we prove the existence of a family of positive radial stable solutions, which satisfies the separation property. We also show the multiple existence of the Joseph--Lundgren critical exponent for some $\ell \in (0,\infty)$ and $s \in (0,1)$, and this property does not hold in the case $s=1$.