Multiscale-bump standing waves with a critical frequency for nonlinear Schrödinger equations
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- by Daomin Cao, Ezzat S. Noussair and Shusen Yan PDF
- Trans. Amer. Math. Soc. 360 (2008), 3813-3837 Request permission
Abstract:
In this paper we study the existence and qualitative property of standing wave solutions $\psi (x,t) = e^{-\frac {iEt}{\hbar }} u(x)$ for the nonlinear Schrödinger equation $i\hbar \frac {\partial \psi }{\partial t} + \frac {\hbar ^2}{2m} \Delta \psi - W(x) \psi + |\psi |^{p-1} \psi = 0$ with $E$ being a critical frequency in the sense that $\inf \limits _{x\in \mathbb {R}^N} W(x)=E.$ We show that if the zero set of $V=W-E$ has $k$ isolated connected components $Z_i (i=1,\cdots , k)$ such that the interior of $Z_i$ is not empty and $\partial Z_i$ is smooth, $V$ has $t$ isolated zero points, $b_i$, $i=1,\cdots ,t$, and $V$ has $l$ critical points $a_i(i=1,\cdots ,l)$ such that $V(a_i)>0$, then for $\hbar > 0$ small, there exists a standing wave solution which is trapped in a neighborhood of $\bigcup _{i=1} Z_i\cup \bigl (\bigcup _{i=1}^t\{b_i\})\cup \bigl (\bigcup _{i=1}^l\{a_i\}\bigr ).$ Moreover the amplitudes of the standing wave around $\bigcup ^k_{i=1} Z_i$, $\bigcup ^t_{i=1}\{b_i\}$ and $\bigcup ^l_{i=1}\{a_i\}$ are of a different order of $\hbar$. This type of multi-scale solution has never before been obtained.References
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Additional Information
- Daomin Cao
- Affiliation: Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China
- MR Author ID: 261647
- Email: dmcao@amt.ac.cn
- Ezzat S. Noussair
- Affiliation: School of Mathematics, The University of New South Wales, Sydney 2052, Australia
- Email: noussair@maths.unsw.edu.au
- Shusen Yan
- Affiliation: School of Mathematics, Statistics and Computer Science, The University of New England, Armidale NSW 2351, Australia
- Email: syan@turing.une.edu.au
- Received by editor(s): May 8, 2006
- Received by editor(s) in revised form: June 15, 2006
- Published electronically: February 13, 2008
- Additional Notes: The first author was supported by the Fund of Distinguished Young Scholars of China and Innovative Funds of CAS in China
The second and third authors were supported by ARC in Australia - © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 360 (2008), 3813-3837
- MSC (2000): Primary 35J20, 35J65
- DOI: https://doi.org/10.1090/S0002-9947-08-04348-1
- MathSciNet review: 2386247