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 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 $\mu >{\mu }_0$.

References in corpus (1)

• Elliptic Equations Involving Meausres