TY - JOUR

T1 - Uniqueness of non-linear ground states for fractional Laplacians in R

AU - Frank, Rupert L.

AU - Lenzmann, Enno

PY - 2013

Y1 - 2013

N2 - We prove uniqueness of ground state solutions Q = Q(|x|) ≥ 0 of the non-linear equation(−Δ)sQ+Q−Qα+1=0inR,where 0 < s < 1 and 0 < α < 4s/(1−2s) for s<12 and 0 < α < ∞ for 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=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)Q α 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.

AB - We prove uniqueness of ground state solutions Q = Q(|x|) ≥ 0 of the non-linear equation(−Δ)sQ+Q−Qα+1=0inR,where 0 < s < 1 and 0 < α < 4s/(1−2s) for s<12 and 0 < α < ∞ for 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=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)Q α 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.

U2 - 10.1007/s11511-013-0095-9

DO - 10.1007/s11511-013-0095-9

M3 - Journal article

VL - 210

SP - 261

EP - 318

JO - Acta Mathematica

JF - Acta Mathematica

SN - 0001-5962

IS - 2

ER -