**Memoirs of the American Mathematical Society**

2013;
89 pp;
Softcover

MSC: Primary 35;
Secondary 58

Print ISBN: 978-0-8218-9163-6

Product Code: MEMO/229/1076

List Price: $71.00

AMS Member Price: $42.60

MAA Member Price: $63.90

**Electronic ISBN: 978-1-4704-1530-3
Product Code: MEMO/229/1076.E**

List Price: $71.00

AMS Member Price: $42.60

MAA Member Price: $63.90

# Semiclassical Standing Waves with Clustering Peaks for Nonlinear Schrödinger Equations

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*Jaeyoung Byeon; Kazunaga Tanaka*

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

# Table of Contents

## Semiclassical Standing Waves with Clustering Peaks for Nonlinear Schrodinger Equations

- Chapter 1. Introduction and results 110 free
- Chapter 2. Preliminaries 918 free
- Chapter 3. Local centers of mass 1524
- Chapter 4. Neighborhood Ω_{𝜀}(𝜌,𝑅,𝛽) and minimization for a tail of 𝑢 in Ω_{𝜀} 2130
- Chapter 5. A gradient estimate for the energy functional 2938
- Chapter 6. Translation flow associated to a gradient flow of 𝑉(𝑥) on \R^{𝑁} 3948
- Chapter 7. Iteration procedure for the gradient flow and the translation flow 4958
- Chapter 8. An (𝑁+1)ℓ₀-dimensional initial path and an intersection result 5362
- Chapter 9. Completion of the proof of Theorem 1.3 6170
- Chapter 10. Proof of Proposition 8.3 6372
- Chapter 11. Proof of Lemma 6.1 7786
- Chapter 12. Generalization to a saddle point setting 8392
- Bibliography 8796