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

Abstract

In this paper, we study the asymptotic behavior of positive solutions of the fractional Hardy-H\'enon equation $$ (-\Delta)^\sigma u = |x|^\alpha u^p ~~~~~~~~~~~ in ~~ B_1 \backslash \{0\} $$ with an isolated singularity at the 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.