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Semiclassical Standing Waves with Clustering Peaks for Nonlinear Schrödinger Equations
 
Jaeyoung Byeon KAIST, Daejeon, Republic of Korea
Kazunaga Tanaka Waseda University, Tokyo, Japan
Front Cover for Semiclassical Standing Waves with Clustering Peaks for Nonlinear Schrodinger Equations
Available Formats:
Electronic ISBN: 978-1-4704-1530-3
Product Code: MEMO/229/1076.E
List Price: $71.00
MAA Member Price: $63.90
AMS Member Price: $42.60
Front Cover for Semiclassical Standing Waves with Clustering Peaks for Nonlinear Schrodinger Equations
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  • Front Cover for Semiclassical Standing Waves with Clustering Peaks for Nonlinear Schrodinger Equations
  • Back Cover for Semiclassical Standing Waves with Clustering Peaks for Nonlinear Schrodinger Equations
Semiclassical Standing Waves with Clustering Peaks for Nonlinear Schrödinger Equations
Jaeyoung Byeon KAIST, Daejeon, Republic of Korea
Kazunaga Tanaka Waseda University, Tokyo, Japan
Available Formats:
Electronic ISBN:  978-1-4704-1530-3
Product Code:  MEMO/229/1076.E
List Price: $71.00
MAA Member Price: $63.90
AMS Member Price: $42.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2292013; 89 pp
    MSC: Primary 35; Secondary 58;

    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
     
     
    • 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
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Volume: 2292013; 89 pp
MSC: Primary 35; Secondary 58;

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 \(f\).

  • 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
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