Removability of singularities and superharmonicity for some fractional Laplacian equations

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