Direct Method of Moving Spheres on Fractional Order Equations

Wenxiong Chen; Yan Li; Ruobing Zhang

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arXiv · 1509.03785

Direct Method of Moving Spheres on Fractional Order Equations

Wenxiong Chen·Yan Li·Ruobing Zhang

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Abstract

In this paper, we introduce a direct method of moving spheres for the nonlocal fractional Laplacian $(-\triangle)^{\alpha/2}$ for $0<\alpha<2$, in which a key ingredient is the narrow region maximum principle. As immediate applications, we classify the non-negative solutions for a semilinear equation involving the fractional Laplacian in $\mathbb{R}^n$; we prove a non-existence result for prescribing $Q_{\alpha}$ curvature equation on $\mathbb{S}^n$; then by combining the direct method of moving planes and moving spheres, we establish a Liouville type theorem on the half Euclidean space. We expect to see more applications of this method to many other equations involving non-local operators.

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Wenxiong Chen, Yan Li, Ruobing Zhang. 2015-09-12. Direct Method of Moving Spheres on Fractional Order Equations. https://arxiv.org/abs/1509.03785

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