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Uniqueness and Nondegeneracy of Ground States for (−Δ)^sQ+Q−Q^(α+1)=0 in R

Frank, Rupert L. and Lenzmann, Enno (2015) Uniqueness and Nondegeneracy of Ground States for (−Δ)^sQ+Q−Q^(α+1)=0 in R. . (Submitted)

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We prove uniqueness of ground state solutions Q = Q(|x|)≥0 for the nonlinear equation (−Δ)^sQ + Q − Q^(α+)1 = 0 in R, where 0 < s < 1 and 0 < α < _(4s) ^(1−2s) for s < 1/2 and 0 < α < ∞ for s ≥ 1/2. Here (−Δ)^s denotes the fractional Laplacian in one dimension. In particular, we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for s = 1/2 and α = 1 in [Acta Math.,167 (1991), 107-126]. As a technical key result in this paper, we show that the associated linearized operator L_+ = (−Δ)^s + 1− (α+1)Q^α is nondegenerate; 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 nonlinear dispersive PDEs with fractional Laplacians, such as the generalized Benjamin-Ono (BO) and Benjamin-Bona-Mahony (BBM) water wave equations.

Item Type:Report or Paper (Discussion Paper)
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Frank, Rupert L.0000-0001-7973-4688
Additional Information:(Submitted on 21 Sep 2010 (v1), last revised 23 Mar 2015 (this version, v2)) R. F. acknowledges support from NSF grant PHY-0652854. E. L. was supported by a Steno fellowship from the Danish science research council, and he also gratefully acknowledges partial support from NSF grant DMS-0702492.
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Danish Natural Science Research CouncilUNSPECIFIED
Record Number:CaltechAUTHORS:20170501-072727175
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:77083
Deposited By: Ruth Sustaita
Deposited On:01 May 2017 16:37
Last Modified:09 Mar 2020 13:18

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