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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
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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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