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

 

Instability of standing waves of the Schrödinger equation with inhomogeneous nonlinearity


Authors: Yue Liu, Xiao-Ping Wang and Ke Wang
Journal: Trans. Amer. Math. Soc. 358 (2006), 2105-2122
MSC (2000): Primary 35B35, 35B60, 35Q35, 35Q40, 35Q55, 76B25, 76E25, 76E30, 78A15
Published electronically: May 9, 2005
MathSciNet review: 2197450
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Abstract | References | Similar Articles | Additional Information

Abstract: This paper is concerned with the inhomogeneous nonlinear Shrödinger equation (INLS-equation)

\begin{displaymath}i u_t + \Delta u + V(\epsilon x) \vert u\vert^p u = 0, \; x \in {\mathbf R}^N. \end{displaymath}

In the critical and supercritical cases $ p \ge 4/N, $ with $ N \ge 2, $ it is shown here that standing-wave solutions of (INLS-equation) on $ H^1({\mathbf R}^N) $ perturbation are nonlinearly unstable or unstable by blow-up under certain conditions on the potential term V with a small $ \epsilon > 0.$


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Additional Information

Yue Liu
Affiliation: Department of Mathematics, University of Texas, Arlington, Texas 76019
Email: yliu@uta.edu

Xiao-Ping Wang
Affiliation: Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
Email: mawang@ust.hk

Ke Wang
Affiliation: California Institute of Technology, MC 217-50, 1200 E. California Boulevard, Pasadena, California 91125
Email: wang@acm.caltech.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03763-3
PII: S 0002-9947(05)03763-3
Keywords: Nonlinear Schr\"odinger equation, inhomogeneous nonlinearities, blow-up, standing waves, ground state, stability theory
Received by editor(s): April 16, 2003
Received by editor(s) in revised form: April 29, 2004
Published electronically: May 9, 2005
Article copyright: © Copyright 2005 American Mathematical Society