Weak and strong singular solutions of semilinear fractional elliptic equations

Let $p\in (0,\frac{N}{N-2\alpha })$, $\alpha \in (0,1)$ and $\mathrm{\Omega }\subset {\mathbb{R}}^N$ 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 $(E_k)$ $(-\mathrm{\Delta }{)}^{\alpha }u+u^p=k{\delta }_0$ in $\mathrm{\Omega }$ which vanishes in ${\mathrm{\Omega }}^c$, is a classical solution of $(E_{\ast })$ $(-\mathrm{\Delta }{)}^{\alpha }u+u^p=0$ in $\mathrm{\Omega }\setminus \{0\}$ with the same outer data. When $\frac{2\alpha }{N-2\alpha }\le 1+\frac{2\alpha }{N}$, $p\in(0, 1+\frac{2\alpha}{N}]$weshowthatthe$u_k$convergesto$\infty$inwhole$\Omega$when$k\to\infty$,while,for$p\in(1+\frac{2\alpha}N,\frac{N}{N-2\alpha})$,thelimitofthe$u_k$ is a strongly singular solution of $(E_{\ast })$. The same result holds in the case $1+\frac{2\alpha }{N}<\frac{2\alpha }{N-2\alpha }$ excepted if $\frac{2\alpha}{N}