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Semiclassical Standing Waves with Clustering Peaks for Nonlinear Schrödinger Equations
Jaeyoung Byeon, KAIST, Daejeon, Republic of Korea, and Kazunaga Tanaka, Waseda University, Tokyo, Japan
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Memoirs of the American Mathematical Society
2013; 89 pp; softcover
Volume: 229
ISBN-10: 0-8218-9163-4
ISBN-13: 978-0-8218-9163-6
List Price: US$71 Individual Members: US$42.60
Institutional Members: US\$56.80
Order Code: MEMO/229/1076

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 $$f$$.

• Introduction and results
• Preliminaries
• Local centers of mass
• Neighborhood $$\Omega_\epsilon(\rho,R,\beta)$$ and minimization for a tail of $$u$$ in $$\Omega_\epsilon$$
• A gradient estimate for the energy functional
• Translation flow associated to a gradient flow of $$V(x)$$ on $${\bf R}^N$$
• Iteration procedure for the gradient flow and the translation flow
• An $$(N+1)\ell_0$$-dimensional initial path and an intersection result
• Completion of the proof of Theorem 1.3
• Proof of Proposition 8.3
• Proof of Lemma 6.1
• Generalization to a saddle point setting
• Bibliography