Direct Method of Moving Spheres on Fractional Order Equations

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In this paper, we introduce a direct method of moving spheres for the nonlocal fractional Laplacian (-)α/² for 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 Rⁿ; we prove a non-existence result for prescribing Qα curvature equation on Sⁿ; 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.