  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
$s=1$.
