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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Spike solutions in coupled nonlinear Schrödinger equations with attractive interaction


Authors: E. N. Dancer and Juncheng Wei
Journal: Trans. Amer. Math. Soc. 361 (2009), 1189-1208
MSC (2000): Primary 35B40, 35B45; Secondary 35J40
Published electronically: October 7, 2008
MathSciNet review: 2457395
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Abstract: We consider the following elliptic system:

$\displaystyle \left\{\begin{array}{l} \varepsilon^2 \Delta u-\lambda_1 u+\mu_1... ... \mbox{in} \ \Omega, \ u=v=0 \ \mbox{on} \ \partial\Omega, \end{array}\right. $

where $ \Omega\subset \mathbb{R}^N (N\leq 3)$ is a smooth and bounded domain, $ \varepsilon>0$ is a small parameter, $ \lambda_1, \lambda_2, \mu_1, \mu_2 >0$ are positive constants and $ \beta \ne 0 $ is a coupling constant. We show that there exists an interval $ I=[a_0, b_0]$ and a sequence of numbers $ 0<\beta_1 <\beta_2 <...<\beta_n <...$ such that for any $ \beta \in (0, +\infty) \backslash (I \cup \{ \beta_1,..., \beta_n, ...\})$, the above problem has a solution such that both $ u$ and $ v$ develop a spike layer at the innermost part of the domain. Central to our analysis is the nondegeneracy of radial solutions in $ \mathbb{R}^N$.


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Additional Information

E. N. Dancer
Affiliation: School of Mathematics and Statistics, University of Sydney, Sydney, Australia
Email: normd@maths.usyd.edu.au

Juncheng Wei
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
Email: wei@math.cuhk.edu.hk

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04735-1
PII: S 0002-9947(08)04735-1
Keywords: Spike layers, Bose-Einstein condensates, coupled nonlinear Schr\"odinger equations
Received by editor(s): August 1, 2006
Published electronically: October 7, 2008
Article copyright: © Copyright 2008 American Mathematical Society