How to Order

For AMS eBook frontlist subscriptions or backfile collection purchases:

   1a. To purchase any ebook backfile or to subscibe to the current year of Contemporary Mathematics, please download this required license agreement,

   1b. To subscribe to the current year of Memoirs of the AMS, please download this required license agreement.

   2. Complete and sign the license agreement.

   3. Email, fax, or send via postal mail to:

Customer Services
American Mathematical Society
201 Charles Street Providence, RI 02904-2213  USA
Phone: 1-800-321-4AMS (4267)
Fax: 1-401-455-4046
Email: cust-serv@ams.org

Visit the AMS Bookstore for individual volume purchases.

Browse the current eBook Collections price list

Powered by MathJax
  Remote Access

Semiclassical standing waves with clustering peaks, for nonlinear Schrödinger equations


About this Title

Jaeyoung Byeon and Kazunaga Tanaka

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: http://dx.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

View full volume PDF

View other years and numbers:

Table of Contents


Chapters

  • Chapter 1. Introduction and results
  • Chapter 2. Preliminaries
  • Chapter 3. Local centers of mass
  • Chapter 4. Neighborhood $\Omega _\varepsilon (\rho ,R,\beta )$ and minimization for a tail of $u$ in $\Omega _\varepsilon $
  • Chapter 5. A gradient estimate for the energy functional
  • Chapter 6. Translation flow associated to a gradient flow of $V(x)$ on ${\bf R}^N$
  • Chapter 7. Iteration procedure for the gradient flow and the translation flow
  • Chapter 8. An $(N+1)\ell _0$-dimensional initial path and an intersection result
  • Chapter 9. Completion of the proof of Theorem 1.3
  • Chapter 10. Proof of Proposition 8.3
  • Chapter 11. Proof of Lemma 6.1
  • Chapter 12. Generalization to a saddle point setting

Abstract


We study the following singularly perturbed problem

Our main result is the existence of a family of solutions with peaks that cluster near a local maximum of . 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 . Earlier works in this direction can be found in [KW, DLY, DY, NY] for ( when , when ). 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 . Such spectral information is difficult to get and can only be obtained for very special . 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 .

References [Enhancements On Off] (What's this?)