Uniqueness of Radial Solutions for the Fractional Laplacian

By Rupert L. Frank, E. Lenzmann, Luis Silvestre. 2013

597 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7ac80a6eecdc63e83467ad8e918545b67a9dcc1f

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Citations in the corpus

Multiplicity and concentration of positive solutions for the fractional Schrödinger–Poisson systems with critical growth (2017, unknown): “See for example [12, 13, 22, 23, 28, 37].”
Ground states for fractional Schrödinger equations with critical growth (2014, background): “Very recently, some papers considered uniqueness and nondegeneracy of ground state solution for the nonlinear equations involving the fractional Laplacian, see [13, 16].”
Concentrating standing waves for the fractional nonlinear Schr\"odinger equation (2013, unknown): “The fractional Laplacian (−∆)su of a function u ∈ H2s(RN ) is defined in terms of its Fourier transform by the relation ̂(−∆)su = |ξ|2sû ∈ L2(RN ).”
An extension problem for the CR fractional Laplacian (2013, unknown): “t contains an energy equality for the fractional norm S˙γ(Hn), using the extension problem from Theorem 1.1. This idea of using the extension has been successfully employed in several other settings ([27, 28, 4], for instance). Proposition 6.1. Let 0 <γ<1 and f∈ S˙γ(Hn). Then Eγf∈ H˙1,γ(Ωn+1) ”
A singularly perturbed Kirchhoff problem revisited (2016, background): “[10, 12, 13, 24] and to [24, 34] for mathematical researches on fractional type nonlocal Schrödinger equations and convolution type nonlocal Choquard equations, repectively.”
Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum (2014, background): “A key ingredient is the linear nondegeneracy of the least energy solution w; this is an important result that was completely achieved only recently in [Frank et al. 2015], after preliminary works in particular cases discussed in [Amick and Toland 1991; Frank and Lenzmann 2013; Fall and Valdinoci 201”
A direct method of moving planes for the fractional Laplacian (2014, background): “Remark 2 In [FLe] and [FLS], Frank, Lenzmann, and Silvestre obtained the radial symmetry and uniqueness of the solution for equation (13) without positivity assumption on u.”
Global uniqueness for the fractional semilinear Schrödinger equation (2017, unknown): “We would like to point out that the uniqueness up to translations of the nontrivial solution of the fractional nonlinear equation holds for certain nonlinearity q(x, u), we refer to [4] and references therein.”
Concentration phenomenon for fractional nonlinear Schr\"{o}dinger equations (2013, background): “ation. The nondegeneracy implies that 0 is an isolated spectral point of L0. More precisely, for all φ∈ (kerL0)⊥, one has kL0φkL2(Rn) ≥ ckφkH2s(Rn) (2.6) for some positive constant c. By Lemma C.2 of [38], it holds that, for j = 1,· · ·,n, ∂ju0 := ∂x j u0 has the following decay estimate, |∂ju0| ≤ C”
ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH THE FRACTIONAL LAPLACIAN (2013, background): “Rn) (1.14) where ǫ>0 is suﬃciently small, 0 <s<1, p∈ (1,n+2s n−2s) and V is a positive bounded C 1,α function whose value is away from 0. In particular, employing the nondegeneracy result of [20], they deduced the existence of various types of spike solutions, like multiple spikes and clust”