Weak and strong singular solutions of semilinear fractional elliptic equations

Abstract

Let $p\in(0,\frac{N}{N-2α})$, $α\in(0,1)$ and $Ω\subset \R^N$ be a bounded $C^2$ domain containing $0$. If $δ_0$ is the Dirac measure at $0$ and $k>0$, we prove that the weakly singular solution $u_k$ of $(E_k)$ $ (-Δ)^αu+u^p=kδ_0 $ in $Ω$ which vanishes in $Ω^c$, is a classical solution of $(E_*)$ $ (-Δ)^αu+u^p=0 $ in $Ω\setminus\{0\}$ with the same outer data. When $\frac{2α}{N-2α}\leq 1+\frac{2α}{N}$, $p\in(0, 1+\frac{2α}{N}]$ we show that the $u_k$ converges to $\infty$ in whole $Ω$ when $k\to\infty$, while, for $p\in(1+\frac{2α}N,\frac{N}{N-2α})$, the limit of the $u_k$ is a strongly singular solution of $(E_*)$. The same result holds in the case $1+\frac{2α}{N}<\frac{2α}{N-2α}$ excepted if $\frac{2α}{N}