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 $(−Δ)^s$ with s∈(0,1) for any space dimensions N≥1. By extending a monotonicity formula found à la Cabré and Sire [9], we show that the linear equation
\[
(−Δ)^su+Vu=0 in \mathbb{R}^N
\]
has at most one radial and bounded solution vanishing at infinity, provided that the potential V is a radial and non-decreasing. 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 $\mathbb{R}^{N+1}_+$, we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation
\[
(−Δ)^sQ+Q−|Q|^αQ=0 in \mathbb{R}^N
\]
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 in [19] and, in particular, the uniqueness result for solitary waves of the Benjamin-Ono equation found by Amick and Toland [4].