The Poisson problem for the fractional Hardy operator: Distributional identities and singular solutions

Abstract: The purpose of this paper is to study and classify singular solutions of the Poisson problem $$ %\begin{equation}\label{eq 0.1} \left { \begin{aligned} {\mathcal L}^s_\mu u = f \quad\ {\rm in}\ \, \Omega\setminus {0},\ u =0 \quad\ {\rm in}\ \, {\mathbb R}^N \setminus \Omega\ %\ %\liminf_{x \to 0}:|u(x)| /\Phi_\mu(x) = k. \end{aligned} \right. $$ for the fractional Hardy operator ${\mathcal L}_\mu^s u= (-\Delta)^s u +\frac{\mu}{|x|^{2s}}u$ in a bounded domain $\Omega \subset {\mathbb R}^N$ ($N \ge 2$) containing the origin. Here $(-\Delta)^s$, $s\in(0,1)$, is the fractional Laplacian of order $2s$, and $\mu \ge \mu_0$, where $\mu_0 = -2^{2s}\frac{\Gamma^2(\frac{N+2s}4)}{\Gamma^2(\frac{N-2s}{4})}<0$ is the best constant in the fractional Hardy inequality. The analysis requires a thorough study of fundamental solutions and associated distributional identities. Special attention will be given to the critical case $\mu= \mu_0$ which requires more subtle estimates than the case $\mu>\mu_0$.