Uniqueness of Radial Solutions for the Fractional Laplacian

We prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional Laplacian (-)(s) with s(0,1) for any space dimensions N1. By extending a monotonicity formula found by Cabre and Sire , we show that the linear equation has at most one radial and bounded solution vanishing at infinity, provided that the potential V is radial and nondecreasing. In particular, this result implies that all radial eigenvalues of the corresponding fractional Schrodinger operator H = (-)(s)+V are simple. Furthermore, by combining these findings on linear equations with topological bounds for a related problem on the upper half-space +N+1, we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation for arbitrary space dimensions N1 and all admissible exponents >0. This generalizes the nondegeneracy and uniqueness result for dimension N = 1 recently obtained by the first two authors and, in particular, the uniqueness result for solitary waves of the Benjamin-Ono equation found by Amick and Toland .(c) 2016 Wiley Periodicals, Inc.