Direct Method of Moving Spheres on Fractional Order Equations
By Wenxiong CHEN *, Yan Li, Ruobing Zhang. 2015
141 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/4a36fccff07bfd4879d6460de1032ae8342ef0cb
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In Applied Mathematics
In Computational Theory and Mathematics
Citations in the corpus
- Asymptotic method of moving planes for fractional parabolic equations (2020, background): “For elliptic equations involving either local or nonlocal operators, a number of systematic approaches have been established to study these qualitative properties, such as the method of moving planes[8, 20, 21, 27, 33, 34, 35, 45], the method of moving spheres[11, 29], and the sliding methods [15, 1”
- Liouville Theorems for Fractional Parabolic Equations (2021, methodology): “To this end, a number of systematic approaches have been established, such as the extension method ([7]), the method of moving planes ([2, 4, 9, 10, 23]), the method of moving spheres ([11, 29]), and the sliding methods ([3, 39]).”