Electronic ISBN:  9781470415303 
Product Code:  MEMO/229/1076.E 
List Price:  $71.00 
MAA Member Price:  $63.90 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 229; 2013; 89 ppMSC: 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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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