Sharp blow up estimates and precise asymptotic behavior of singular positive solutions to fractional Hardy-H\'enon equations

In this paper, we study the asymptotic behavior of positive solutions of the fractional Hardy-H'enon equation $$ (-\Delta)^\sigma u = |x|^\alpha u^p

origin, where $\sigma \in (0, 1)$ and the punctured unit ball $B_1 \backslash
\{0\} \subset \mathbb{R}^n$ with $n \geq 2$. When $-2\sigma < \alpha < 2\sigma$
and $\frac{n+\alpha}{n-2\sigma} < p < \frac{n+2\sigma}{n-2\sigma}$, we give a
classification of isolated singularities of positive solutions, and in
particular, this implies sharp blow up estimates of singular solutions.
Further, we describe the precise asymptotic behavior of solutions near the
singularity. More generally, we classify isolated boundary singularities and
describe the precise asymptotic behavior of singular solutions for a relevant
degenerate elliptic equation with a nonlinear Neumann boundary condition. These
results parallel those known for the Laplacian counterpart proved by Gidas and
Spruck (Comm. Pure Appl. Math. 34: 525-598, 1981), but the methods are very
different, since the ODEs analysis is a missing ingredient in the fractional
case. Our proofs are based on a monotonicity formula, combined with blow up
(down) arguments, Kelvin transformation and uniqueness of solutions of related
degenerate equations on $\mathbb{S}^{n}_+$. We also investigate isolated
singularities located at infinity of fractional Hardy-H\'enon equations.