On Isolated Singularities of Fractional Semi-Linear Elliptic Equations

Hui Yang; Wenming Zou

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arXiv · 1804.00817

On Isolated Singularities of Fractional Semi-Linear Elliptic Equations

Hui Yang·Wenming Zou

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Abstract

In this paper, we study the local behavior of nonnegative solutions of fractional semi-linear equations $(-\Delta)^\sigma u = u^p$ 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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Hui Yang, Wenming Zou. 2018-04-03. On Isolated Singularities of Fractional Semi-Linear Elliptic Equations. https://arxiv.org/abs/1804.00817

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