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

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