Remarks on weak solutions of fractional elliptic equations

Abstract

In this note, we continue our study of weak solution $u_k$ to fractional elliptic equation $(-\Delta)^\alpha u+u^p=k\delta_0$ in $\Omega$ which vanishes in $\Omega^c$, where $\Omega\subset \mathbb{R}^N (N\ge2)$ is an open $C^2$ domain containing $0$, $(-\Delta)^\alpha$ with $\alpha\in(0,1)$ is the fractional Laplacian, $k>0$ and $\delta_0$ is the Dirac mass at $0$. We prove that the limit of $u_k$ as $k\to\infty$ blows up in whole $\Omega$ when $p=\min\{1+\frac{2\alpha}{N},\frac{N}{2\alpha}\}$ and $1+\frac{2\alpha}{N}\not=\frac{N}{2\alpha}$.