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

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({pipe}x{pipe}) ≥ 0 of the non-linear equation (-Δ)s Q+Q-Qα+1= 0 in ℝ,(-Δ)s Q + Q - Q α + = 0 in ℝ where 0 < s < 1 and 0 < α < 4s/(1-2s) for s < 1/2 s < and 0 < α < ∞ for s ≥ 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 = 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.

AB - We prove uniqueness of ground state solutions Q = Q({pipe}x{pipe}) ≥ 0 of the non-linear equation (-Δ)s Q+Q-Qα+1= 0 in ℝ,(-Δ)s Q + Q - Q α + = 0 in ℝ where 0 < s < 1 and 0 < α < 4s/(1-2s) for s < 1/2 s < and 0 < α < ∞ for s ≥ 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 = 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.

U2 - 10.1007/s11511-013-0095-9

DO - 10.1007/s11511-013-0095-9

M3 - Journal article

SN - 0001-5962

VL - 210

SP - 261

EP - 318

JO - Acta Mathematica

JF - Acta Mathematica

IS - 2

ER -