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2006 On multi-bump semi-classical bound states of nonlinear Schrödinger equations with electromagnetic fields
Thomas Bartsch, E. Norman Dancer, Shuangjie Peng
Adv. Differential Equations 11(7): 781-812 (2006).

Abstract

We consider the existence and asymptotic behavior of standing wave solutions to nonlinear Schrödinger equations with electromagnetic fields: $ih\frac{\partial\psi}{\partial t} =\left(\frac{h}{i}\nabla-A(x)\right)^2\psi+W(x)\psi-f(|\psi|^2)\psi$ on ${{\mathbb R}}\times{\Omega}$. $\Omega\subset{{\mathbb R}}^N$ is a domain which may be bounded or unbounded. For $h>0$ small we obtain the existence of multi-bump bound states $\psi_h (x,t)=e^{-iEt/h}u_h(x)$ where $u_h$ concentrates simultaneously at possibly degenerate, non-isolated local minima of $W$ as $h\to0$. We require that $W\geq E$ and allow the possibility that $\{x\in\Omega:W(x)=E\}\not=\emptyset$. Moreover, we describe the asymptotic behavior of $u_h$ as $h\to\, 0$.

Citation

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Thomas Bartsch. E. Norman Dancer. Shuangjie Peng. "On multi-bump semi-classical bound states of nonlinear Schrödinger equations with electromagnetic fields." Adv. Differential Equations 11 (7) 781 - 812, 2006.

Information

Published: 2006
First available in Project Euclid: 18 December 2012

zbMATH: 1146.35081
MathSciNet: MR2236582

Subjects:
Primary: 35J60
Secondary: 35B33, 35Q55

Rights: Copyright © 2006 Khayyam Publishing, Inc.

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