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Semiclassical Standing Waves with Clustering Peaks for Nonlinear Schrödinger Equations
Jaeyoung Byeon, KAIST, Daejeon, Republic of Korea, and Kazunaga Tanaka, Waseda University, Tokyo, Japan

Memoirs of the American Mathematical Society
2013; 89 pp; softcover
Volume: 229
ISBN-10: 0-8218-9163-4
ISBN-13: 978-0-8218-9163-6
List Price: US$71
Individual Members: US$42.60
Institutional Members: US$56.80
Order Code: MEMO/229/1076
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The authors study the following singularly perturbed problem: \(-\epsilon^2\Delta u+V(x)u = f(u)\) in \(\mathbf{R}^N\). Their 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 \(f\).

Table of Contents

  • Introduction and results
  • Preliminaries
  • Local centers of mass
  • Neighborhood \(\Omega_\epsilon(\rho,R,\beta)\) and minimization for a tail of \(u\) in \(\Omega_\epsilon\)
  • A gradient estimate for the energy functional
  • Translation flow associated to a gradient flow of \(V(x)\) on \({\bf R}^N\)
  • Iteration procedure for the gradient flow and the translation flow
  • An \((N+1)\ell_0\)-dimensional initial path and an intersection result
  • Completion of the proof of Theorem 1.3
  • Proof of Proposition 8.3
  • Proof of Lemma 6.1
  • Generalization to a saddle point setting
  • Bibliography
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