A direct method of moving spheres on fractional order equations

摘要

In this paper, we introduce a direct method of moving spheres for the fractional Laplacian (-Delta)alpha/2 with 0 < alpha < 2, in which a key ingredient is the narrow region maximum principle. As immediate applications, we classify non-negative solutions for semilinear equations involving the fractional Laplacian in Rn; we prove a non-existence result for the prescribing Q(alpha) curvature equation on S-n; then by combining the direct method of moving planes and moving spheres, we establish a Lionville type theorem on a half Euclidean space. We expect to see more applications of this method to many other nonlinear equations involving non-local operators. (C) 2017 Elsevier Inc. All rights reserved.