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 (−∆) with s ∈ (0, 1) for any space dimensions N > 1. By extending a monotonicity formula found by Cabré and Sire [9], we show that the linear equation (−∆)u+ V u = 0 in R has at most one radial and bounded solution vanishing at infinity, provided that the potential V is radial and non-decreasing. In particular, this result implies that all radial eigenvalues of the corresponding fractional Schrödinger operator H = (−∆) + V are simple. Furthermore, by combining these findings on linear equations with topological bounds for a related problem on the upper half-space R + , we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation (−∆)Q+Q− |Q|Q = 0 in R for arbitrary space dimensions N > 1 and all admissible exponents α > 0. This generalizes the nondegeneracy and uniqueness result for dimension N = 1 recently obtained by the first two authors in [20] and, in particular, the uniqueness result for solitary waves of the Benjamin–Ono equation found by Amick and Toland [4].