Classification of isolated singularities of nonnegative solutions to fractional semi-linear elliptic equations and the existence results

In this paper, we classify the singularities of nonnegative solutions to fractional elliptic equation \begin{equation}\label{eq 0.1} \arraycolsep=1pt \begin{array}{lll} \displaystyle (-\Delta)^\alpha u=u^p\quad &{\rm in}\quad \Omega\setminus{0},\[2mm] \phantom{ (-\Delta)^\alpha } \displaystyle u=0\quad &{\rm in}\quad $\mathbb{R}$^N\setminus\Omega, \end{array} \end{equation} where $p>1$, $\mathrm{\Omega }$ is a bounded, $C^2$ domain in ${\mathbb{R}}^N$ containing the origin, $N\ge 2$ and the fractional Laplacian $(-\mathrm{\Delta }{)}^{\alpha }$ is defined in the principle value sense. We obtain that any classical solution $u$ of (\ref{eq 0.1}) is a weak solution of \begin{equation}\label{eq 0.2} \arraycolsep=1pt \begin{array}{lll} \displaystyle (-\Delta)^\alpha u=u^p+k\delta_0\quad &{\rm in}\quad \Omega,\[2mm] \phantom{ (-\Delta)^\alpha } \displaystyle u=0\quad &{\rm in}\quad $\mathbb{R}$^N\setminus\Omega \end{array} \end{equation} for some $k\ge 0$, where ${\delta }_0$ is the Dirac mass at the origin. In particular, when $p\ge \frac{N}{N-2\alpha }$, we have that $k=0$; when $p<\frac{N}{N-2\alpha }$, $u$ has removable singularity at the origin if $k=0$ and if $k>0$, $u$ satisfies $$\underset{x\to 0}{\lim }u(x)\mathrm{\mid }x{\mathrm{\mid }}^{N-2\alpha }=c_{N,\alpha }k,$$ where $c_{N,\alpha }>0$. Furthermore, when $p\in (1,\frac{N}{N-2\alpha })$, we obtain that there exists $k^{\ast }>0$ such that problem (\ref{eq 0.1}) has at least two positive solutions for $k<k^{\ast }$, a unique positive solution for $k=k^{\ast }$ and no positive solution for $k>k^{\ast }$.