| eBook 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 |
| eBook 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 |
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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 \(f\).
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Table of Contents
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Chapters
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1. Introduction and results
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2. Preliminaries
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3. Local centers of mass
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4. Neighborhood $\Omega _\varepsilon (\rho ,R,\beta )$ and minimization for a tail of $u$ in $\Omega _\varepsilon $
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5. A gradient estimate for the energy functional
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6. Translation flow associated to a gradient flow of $V(x)$ on $\mathbf {R}^N$
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7. Iteration procedure for the gradient flow and the translation flow
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8. An $(N+1)\ell _0$-dimensional initial path and an intersection result
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9. Completion of the proof of Theorem
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10. Proof of Proposition
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11. Proof of Lemma
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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
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10. Proof of Proposition
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11. Proof of Lemma
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12. Generalization to a saddle point setting
