Multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations |
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Authors: | Daomin Cao Ezzat S Noussair |
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Affiliation: | a Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, PR China b School of Mathematics, The University of New South Wales, Sydney 2052, Australia |
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Abstract: | In this paper we study the existence and qualitative property of standing wave solutions for the nonlinear Schrödinger equation with E being a critical frequency in the sense that . We show that if the zero set of W−E has several isolated connected components Zi(i=1,…,m) such that the interior of Zi is not empty and ∂Zi is smooth, then for ?>0 small there exists, for any integer k,1?k?m, a standing wave solution which is trapped in a neighborhood of , where is any given subset of . Moreover the amplitude of the standing wave is of the level . This extends the result of Byeon and Wang (Arch. Rational Mech. Anal. 165 (2002) 295) and is in striking contrast with the non-critical frequency case , which has been studied extensively in the past 20 years. |
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Keywords: | Multi-bump Standing waves Variational method Nonlinear Schrö dinger equation |
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