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

TL;DR

This paper investigates singular solutions to a fractional Poisson problem involving the Hardy operator, providing classification, distributional identities, and analysis of fundamental solutions, especially focusing on the critical Hardy constant case.

Contribution

It introduces a detailed classification of singular solutions for the fractional Hardy operator Poisson problem, including distributional identities and fundamental solutions, with special emphasis on the critical case.

Findings

Classification of singular solutions in the subcritical and critical cases

Distributional identities for the fractional Hardy operator

Explicit fundamental solutions and their properties

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 $\mu>\mu_0$.