Removability of singularities and superharmonicity for some fractional Laplacian equations

Abstract

We study some qualitative properties (including removable singularities and superharmonicity) of non-negative solutions to $$ (-\Delta)^\gamma u=fu^p\quad\text{in }\mathbb R^n\setminus\Sigma $$ which are singular at $\Sigma$. Here $\gamma \in (0, \frac{n}{2})$. Among other things, we first prove that if $\Sigma$ is a compact set in $\mathbb R^n$ with Assouad dimension $\bf d$ (not necessarily an integer), ${\bf d}<n-2\gamma$, and $ u\in L_\gamma(\mathbb R^n)\cap L^p_{loc}({\mathbb R^n\setminus\Sigma})$ is a non-negative solution for some $$p>\frac{n-\bf d}{n-{\bf d}-2\gamma},$$ then $u\in L^p_{loc}(\mathbb R^n)$ and $u$ is a distributional solution in $\mathbb R^n$. Then we prove that $ (-\Delta)^\sigma u >0$ for all $ \sigma \in (0, \gamma)$, if $\Sigma=\phi$.