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Semiclassical standing waves with clustering peaks, for nonlinear Schrödinger equations
About this Title
Jaeyoung Byeon, Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 305-701, Republic of Korea and Kazunaga Tanaka, Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan
Publication: Memoirs of the American Mathematical Society
Publication Year:
2014; Volume 229, Number 1076
ISBNs: 978-0-8218-9163-6 (print); 978-1-4704-1530-3 (online)
DOI: https://doi.org/10.1090/memo/1076
Published electronically: October 10, 2013
Keywords: Nonlinear Schödinger equations,
Singular perturbation,
semi-classical standing waves,
local variational method,
interaction estimate,
translation flow
MSC: Primary 35J60, (35B25, 35Q55, 58E05)
Table of Contents
Chapters
- 1. Introduction and results
- 2. Preliminaries
- 3. Local centers of mass
- 4. Neighborhood $\Omega _\varepsilon (\rho ,R,\beta )$ and minimization for a tail of $u$ in $\Omega _\varepsilon$
- 5. A gradient estimate for the energy functional
- 6. Translation flow associated to a gradient flow of $V(x)$ on $\mathbf {R}^N$
- 7. Iteration procedure for the gradient flow and the translation flow
- 8. An $(N+1)\ell _0$-dimensional initial path and an intersection result
- 9. Completion of the proof of Theorem
- 10. Proof of Proposition
- 11. Proof of Lemma
- 12. Generalization to a saddle point setting
Abstract
We study the following singularly perturbed problem \[ -\epsilon ^2\Delta u+V(x)u = f(u) \quad \hbox {in}\ \mathbf {R}^N. \] Our main result is the existence of a family of solutions with peaks that cluster near a local maximum of $V(x)$. A local variational and deformation argument in an infinite dimensional space is developed to establish the existence of such a family for a general class of nonlinearities $f$. Earlier works in this direction can be found in [KW, DLY, DY, NY] for $f(\xi )=\xi ^p$ ($1 < p < \frac { N+2}{N-2}$ when $N \geq 3$, $1< p < \infty$ when $N=1,2$). These papers use the Lyapunov-Schmidt reduction method, where it is essential to have information about the null space of the linearization of a solution of the limit equation $-\Delta u+u =u^p$. Such spectral information is difficult to get and can only be obtained for very special $f’s$. Our new approach in this memoir does not require such a detailed knowledge of the spectrum and works for a much more general class of nonlinearities $f$.- A. Ambrosetti, M. Badiale, and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal. 140 (1997), no. 3, 285–300. MR 1486895, DOI 10.1007/s002050050067
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