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December  2009, 25(4): 1229-1247. doi: 10.3934/dcds.2009.25.1229

Existence and stability of high frequency standing waves for a nonlinear Schrödinger equation

François Genoud 1,

1. 

IACS-FSB, Section de Mathématiques, École Polytechnique Fédérale de Lausanne CH-1015 Lausanne, Switzerland

Received  November 2008 Revised  April 2009 Published  September 2009

This article is concerned with the existence and orbital stability of standing waves for a nonlinear Schrödinger equation (NLS) with a nonautonomous nonlinearity. It continues and concludes the series of papers [6, 7, 8]. In [6], the authors make use of a continuation argument to establish the existence in $R \times H^1$(RN$)$ of a smooth local branch of solutions to the stationary elliptic problem associated with (NLS) and hence the existence of standing wave solutions of (NLS) with small frequencies. Complementary conditions on the nonlinearity are found, under which either stability of the standing waves and bifurcation of the branch of solutions from the point $(0,0)\in R \times H^1$(RN$)$ occur, or instability and asymptotic bifurcation occur. The main hypotheses in [6] concern the behaviour of the nonlinearity with respect to the space variable at infinity. The paper [7] extends the results of [6] to (NLS) with more general nonlinearities. In [8], the global continuation of the local branch obtained in [6] is proved under additional hypotheses on the nonlinearity. In particular, spherical symmetry with respect to the space variable is assumed. The aim of the present work is to prove the existence and discuss the orbital stability of standing waves with high frequencies, independently of the results obtained in [6] and [8]. The main hypotheses now concern the behaviour of the nonlinearity with respect to the space variable around the origin. The methods are the same in spirit as that of [6] and permit to discuss the asymptotic behaviour of the global branch of solutions obtained in [8].
Keywords: standing waves, orbital stability., Bifurcation.
Mathematics Subject Classification: Primary: 35J60, 35B32; Secondary: 37C7.
Citation: François Genoud. Existence and stability of high frequency standing waves for a nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1229-1247. doi: 10.3934/dcds.2009.25.1229
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