Removability of singularities and superharmonicity for some fractional Laplacian equations
Weiwei Ao, María del Mar González, Ali Hyder, Juncheng Wei · arXiv (Cornell University) · 2020
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}\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 $Σ=ϕ$.