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

Abstract

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 $p>1$, $\Omega$ is a bounded, $C^2$ domain in $\mathbb{R}^N$ containing the origin, $N\ge2$ and the fractional Laplacian $(-\Delta)^\alpha$ 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 $k\ge0$, 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 $$\lim_{x\to0} u(x)|x|^{N-2\alpha}=c_{N,\alpha}k,$$ where $c_{N,\alpha}>0$. Furthermore, when $p\in(1, \frac{N}{N-2\alpha})$, we obtain that there exists $k^*>0$ such that problem (\ref{eq 0.1}) has at least two positive solutions for $k<k^*$, a unique positive solution for $k=k^*$ and no positive solution for $k>k^*$.