Removability of singularities and superharmonicity for some fractional Laplacian equations
英文摘要：
We study some qualitative properties (including removable singularities and superharmonicity) of non-negative solutions to $$ (-Δ)^γu=fu^p\quad\text{in }\mathbb R^n\setminusΣ$$ which are singular at $Σ$. Here $γ\in (0, \frac{n}{2})$. Among other things, we first prove that if $Σ$ is a compact set in $\mathbb R^n$ with Assouad dimension $\bf d$ (not necessarily an integer), ${\bf d}<n-2γ$, and $ u\in L_γ(\mathbb R^n)\cap L^p_{loc}({\mathbb R^n\setminusΣ})$ is a non-negative solution for some $$p>\frac{n-\bf d}{n-{\bf d}-2γ},$$ then $u\in L^p_{loc}(\mathbb R^n)$ and $u$ is a distributional solution in $\mathbb R^n$. Then we prove that $ (-Δ)^σu >0$ for all $ σ\in (0, γ)$, if $Σ=ϕ$.
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