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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)
Published electronically: October 10, 2013
Keywords:Nonlinear Schödinger equations, Singular perturbation, semi-classical standing waves, local variational method, interaction estimate

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Table of Contents


  • 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


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 .

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