# Uniqueness of Radial Solutions for the Fractional Laplacian

Frank, Rupert L. and Lenzmann, Enno and Silvestre, Luis. (2013) Uniqueness of Radial Solutions for the Fractional Laplacian. Preprints Fachbereich Mathematik, 2013 (06).

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## 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 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].
Faculties and Departments: 05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Analysis (Lenzmann)12 Special Collections > Preprints Fachbereich Mathematik Lenzmann, Enno Preprint Universität Basel English 12 May 2019 23:27 28 Mar 2019 09:52

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