Uniqueness and Nondegeneracy of Ground States for $(-Δ)^s Q + Q - Q^{α+1} = 0$ in $\R$

Mathematics – Analysis of PDEs

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

45 pages

Scientific paper

We prove uniqueness of ground state solutions $Q = Q(|x|) \geq 0$ for the nonlinear equation $(-\Delta)^s Q + Q - Q^{\alpha+1}= 0$ in $\R$, where $0 < s < 1$ and $0 < \alpha < \frac{4s}{1-2s}$ for $s < 1/2$ and $0 < \alpha < \infty$ for $s \geq 1/2$. Here $(-\Delta)^s$ 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 $\alpha=1$ in [Acta Math., \textbf{167} (1991), 107--126]. As a technical key result in this paper, we show that the associated linearized operator $L_+ = (-\Delta)^s + 1 - (\alpha+1) Q^\alpha$ is nondegenerate; i.\,e., its kernel satisfies $\mathrm{ker}\, L_+ = \mathrm{span} \, \{ Q' \}$. 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.

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

Rating

Uniqueness and Nondegeneracy of Ground States for $(-Δ)^s Q + Q - Q^{α+1} = 0$ in $\R$ does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Uniqueness and Nondegeneracy of Ground States for $(-Δ)^s Q + Q - Q^{α+1} = 0$ in $\R$, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Uniqueness and Nondegeneracy of Ground States for $(-Δ)^s Q + Q - Q^{α+1} = 0$ in $\R$ will most certainly appreciate the feedback.

Rate now

     

Profile ID: LFWR-SCP-O-697412

  Search
All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.