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 $$(-\hat{I}”{)}^{3}u=f{u}^p\text{in }{\mathbb{R}}^n\setminus \hat{I}\pounds$$ which are singular at $\hat{I}\pounds$. Here ${\hat{I}}^3\in (0,\frac{n}{2})$. Among other things, we first prove that if $\hat{I}\pounds$ is a compact set in ${\mathbb{R}}^n$ with Assouad dimension $\mathbf{d}$ (not necessarily an integer), $\mathbf{d}<n-2{\hat{I}}^3$, and $u\in L_{\hat{I}}^3({\mathbb{R}}^n)\cap L_{loc}^p({\mathbb{R}}^n\setminus \hat{I}\pounds )$ is a non-negative solution for some $$p>\frac{n-\mathbf{d}}{n-\mathbf{d}-2{\hat{I}}^3},$$ then $u\in L_{loc}^p({\mathbb{R}}^n)$ and $u$ is a distributional solution in ${\mathbb{R}}^n$. Then we prove that $(-\hat{I}”{)}^{\ddot{I}}ƒu>0$ for all $\ddot{I}ƒ\in (0,{\hat{I}}^3)$, if $\hat{I}\pounds =\ddot{I}•$.

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