No abstract

Let n be a positive integer and let 0 < α < n. In this paper, we continue our study of the integral equationWe mainly consider singular solutions in subcritical, critical, and super critical cases, and obtain qualitative properties, such as radial symmetry, monotonicity, and upper bounds for the solutions.

ABSTRACT. In this paper, we introduce a direct method of moving spheres for the nonlocal fractional Laplacian (−△) α/2 with 0 < α < 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 R n ; we prove a non-existence result for prescribing Q α curvature equation on S n ; 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.

No abstract