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)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.

An extension problem for the CR fractional Laplacian

“…This idea of using the extension has been successfully employed in several other settings ( [27,28,4], for instance).…”

Remarks on the inhomogeneous fractional nonlinear Schrödinger equation

“…When γ = 0 and α ∈ (0, 1), the problem (1.1) is a nonlocal model known as nonlinear fractional Schrödinger equation which has also attracted much attentions recently [9,10,11,12,19,20,21,22,23,24,15,16]. The fractional Schrödinger equation is a fundamental equation of fractional quantum mechanics, which was derived by Laskin [29,30] as a result of extending the Feynman path integral, from the Brownian-like to Levy-like quantum mechanical paths.…”

Existence and mass concentration of pseudo-relativistic Hartree equation

“…The optimal constant in inequality (1.11) was determined in [8]. Taking into account as [20] the Weinstein functional 12) we know from Lemma 5 in [18] that 13) and then, the best constant C gn in (1.11) is equal to 2 Q 2 2 . Our first result is as follows.…”