In this paper, we study the local behavior of nonnegative solutions of fractional semi-linear equations ${(-\mathrm{\Delta })}^{\sigma }u=u^p$ with an isolated singularity, where $\sigma \in (0,1)$ and $\frac{n}{n-2\sigma }<p<\frac{n+2\sigma }{n-2\sigma }$. We first use the blow up method and a Liouville type theorem to derive an upper bound. Then we establish a monotonicity formula and a sufficient condition for removable singularity to give a classification of the isolated singularities. When $\sigma =1$, this classification result has been proved by Gidas and Spruck (1981) [23], Caffarelli et al. (1989) [7].

中文翻译：

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分数阶半线性椭圆型方程的孤立奇点

在本文中，我们研究分数半线性方程的非负解的局部性质 ${（-\mathrm{\Delta }）}^{\sigma }ü={ü}^p$ 具有孤立的奇点， $\sigma \in （0，\mathrm{1个}）$ 和 $\frac{ñ}{ñ-2\sigma }<p<\frac{ñ+2\sigma }{ñ-2\sigma }$。我们首先使用爆炸方法和Liouville型定理来推导上限。然后，我们建立了单调性公式和可移动奇点的充分条件，以给出孤立奇点的分类。什么时候$\sigma =\mathrm{1个}$，这种分类结果已经由Gidas和Spruck（1981）[23]，Caffarelli等人证明。（1989）[7]。