A direct method of moving spheres on fractional order equations

In this paper, we introduce a direct method of moving spheres for the fractional Laplacian (−△)^α/2 with 0>α>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_α curvature equation on Sⁿ; then by combining the direct method of moving planes and moving spheres, we establish a Liouville 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.