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On Isolated Singularities of Fractional Semi-Linear Elliptic Equations (1804.00817v1)

Published 3 Apr 2018 in math.AP

Abstract: In this paper, we study the local behavior of nonnegative solutions of fractional semi-linear equations $(-\Delta)\sigma u = up$ with an isolated singularity, where $\sg \in (0, 1)$ and $\frac{n}{n-2\sg} < p < \frac{n+2\sg}{n-2\sg}$. We first use 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 $\sg=1$, this classification result has been proved by Gidas and Spruck (Comm. Pure Appl. Math. 34: 525-598, 1981).

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