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 $\Omega \subset {\mathbb{R}}^N$ 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 $(E_k)$ $(-\Delta {)}^{\alpha }u+u^p=k{\delta }_0$ in $\Omega$ which vanishes in ${\Omega }^c$, is a classical solution of $(E_{\ast })$ $(-\Delta {)}^{\alpha }u+u^p=0$ in $\Omega \setminus \{0\}$ with the same outer data. When $\frac{2\alpha }{N-2\alpha }\le 1+\frac{2\alpha }{N}$, $p\in (0,1+\frac{2\alpha }{N}]$ 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 $(E_{\ast })$. The same result holds in the case $1+\frac{2\alpha }{N}<\frac{2\alpha }{N-2\alpha }$ excepted if $\frac{2α}{N}

A paraître, Asymptotic Analysis

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