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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
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 Click above image for expanded view 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 $f$.

• 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
• Requests

Review Copy – for reviewers who would like to review an AMS book
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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
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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