On weak solutions to a fractional Hardy-Hénon equation: Part II: Existence

Abstract

This paper and [29] treat the existence and nonexistence of stable weak solutions to a fractional Hardy--Hénon equation $(-\Delta {)}^{s}u=|x{|}^{\ell }|u{|}^{p-1}u$ in ${\mathbb{R}}^N$, where $0<s<1$, $\ell >-2s$, $p>1$, $N\ge 1$ 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 $s=1$.

52 pages