Uniqueness of Radial Solutions for the Fractional Laplacian

• Published in: Communications on pure and applied mathematics Vol. 69; no. 9; pp. 1671 - 1726
• Main Authors: Frank, Rupert L., Lenzmann, Enno, Silvestre, Luis
• Format: Journal Article
• Language: English
• Published: New York Blackwell Publishing Ltd 01.09.2016; John Wiley and Sons, Limited
• Subjects: Laplace transforms; Linear equations; Nonlinear equations; Schrodinger equation
• ISSN: 0010-3640; 1097-0312
• DOI: 10.1002/cpa.21591

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)su+Vu=0 in ℝN 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)sQ+Q−|Q|αQ=0 in ℝN 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 .© 2016 Wiley Periodicals, Inc.