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*} \left \{ \begin {aligned} \mathcal {L}^s_\mu u = f \quad \ \text {in}\ \, \Omega \setminus \{0\},\\ u =0 \quad \ \text {in}\ \, \mathbb {R}^N \setminus \Omega \ \end{aligned} \right . \end{equation*} 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 $\mu >\mu _0$.