Direct Method of Moving Spheres on Fractional Order Equations

Abstract

In this paper, we introduce a direct method of moving spheres for the nonlocal fractional Laplacian $(-△{)}^{\alpha /2}$ 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.

References in corpus (3)

• On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions
• On a fractional Nirenberg problem, part II: existence of solutions
• Symmetry via antisymmetric maximum principles in nonlocal problems of variable order

Cited by in corpus (3)

• Classification of nonnegative solutions to static Schrödinger-Hartree and Schrödinger-Maxwell equations with combined nonlinearities
• Semilinear integro-differential equations, II: one-dimensional and saddle-shaped solutions to the Allen-Cahn equation
• Delaunay-type singular solutions for the fractional Yamabe Problem