Uniqueness of Radial Solutions for the Fractional Laplacian

Abstract

We prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional Laplacian (−Δ)^s with s ∊ (0,1) for any space dimensions N ≥ 1. By extending a monotonicity formula found by Cabré and Sire , we show that the linear equation (−Δ)^s u + Vu=0 in R^N 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 Schrödinger operator H = (−Δ)^s + V are simple. Furthermore, by combining these findings on linear equations with topological bounds for a related problem on the upper half-space R^N_+^(+1), we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation (−Δ)^s Q + Q- │Q│^ɑ Q=0 in R^N 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 and, in particular, the uniqueness result for solitary waves of the Benjamin-Ono equation found by Amick and Toland.

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