Uniqueness of non-linear ground states for fractional Laplacians in $${\mathbb{R}}$$R

By Rupert L. Frank, E. Lenzmann. 2010

338 citations recorded by the source corpus.

AbstractWe prove uniqueness of ground state solutions Q = Q(|x|) ≥ 0 of the non-linear equation $$(-\Delta)^s Q+Q-Q^{\alpha+1}= 0 \quad {\rm in} \, \mathbb{R},$$(−Δ)sQ+Q−Qα+1=0inR,where 0 < s < 1 and 0 < α < 4s/(1−2s) for $${s<\frac{1}{2}}$$s<12 and 0 < α < ∞ for $${s\geq \frac{1}{2}}$$s≥12. Here (−Δ)s denotes the fractional Laplacian in one dimension. In particular, we answer affirmatively an open question recently raised by Kenig–Martel–Robbiano and we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for $${s=\frac{1}{2}}$$s=12 and α = 1 in [5] for the Benjamin–Ono equation.As a technical key result in this paper, we show that the associated linearized operator L+ = (−Δ)s+1−(α+1)Qα is non-degenerate; i.e., its kernel satisfies ker L+ = span{Q′}. This result about L+ proves a spectral assumption, which plays a central role for the stability of solitary waves and blowup analysis for non-linear dispersive PDEs with fractional Laplacians, such as the generalized Benjamin–Ono (BO) and Benjamin–Bona–Mahony (BBM) water wave equations.

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

View Paper: https://www.semanticscholar.org/paper/4de8cbe4b80c014951de7a520301847d43f9f7e1

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In Applied Mathematics

In Control and Systems Engineering

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Citations in the corpus

An extension problem for the CR fractional Laplacian (2013, positive): “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) ”
Nondegeneracy of the bubble in the critical case for nonlocal equations (2013, methodology): “[8] R.L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in R. Acta Math.”
Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum (2014, background): “See [Amick and Toland 1991; Frank and Lenzmann 2013; Fall and Valdinoci 2014] for the (delicate) proof of uniqueness in special situations and [Frank et al.”