Weak and strong singular solutions of semilinear fractional elliptic equations

Abstract

Let $p\in(0,\frac{N}{N-2\alpha})$, $\alpha\in(0,1)$ and $\Omega\subset \R^N$ be a bounded $C^2$ domain containing $0$. If $\delta_0$ is the Dirac measure at $0$ and $k>0$, we prove that the weakly singular solution $u_k$ of $(E_k)$ $ (-\Delta)^\alpha u+u^p=k\delta_0 $ in $\Omega$ which vanishes in $\Omega^c$, is a classical solution of $(E_*)$ $ (-\Delta)^\alpha u+u^p=0 $ in $\Omega\setminus\{0\}$ with the same outer data. When $\frac{2\alpha}{N-2\alpha}\leq 1+\frac{2\alpha}{N}$, $p\in(0, 1+\frac{2\alpha}{N}]$ we show that the $u_k$ converges to $\infty$ in whole $\Omega$ when $k\to\infty$, while, for $p\in(1+\frac{2\alpha}N,\frac{N}{N-2\alpha})$, the limit of the $u_k$ is a strongly singular solution of $(E_*)$. The same result holds in the case $1+\frac{2\alpha}{N}<\frac{2\alpha}{N-2\alpha}$ excepted if $\frac{2\alpha}{N}