Existence of unstable standing waves for the inhomogeneous nonlinear Schrödinger equation

Yue Liu

doi:10.3934/cpaa.2008.7.193

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2008, Volume 7, Issue 1: 193-209. Doi: 10.3934/cpaa.2008.7.193

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Existence of unstable standing waves for the inhomogeneous nonlinear Schrödinger equation

Yue Liu^1,

1.
Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019-0408

Received: December 31, 2006

Revised: May 31, 2007

Published: December 2007

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Abstract

We establish a sharp instability theorem for the standing-wave solutions of the inhomogeneous nonlinear Schrödinger equation

$i u_t + \Delta u + V(\epsilon x ) |u|^{p-1} u = 0, \quad x \in \mathbf R^n$

with the critical power $ p = 1 + 4/n, n \ge 2, $ under certain conditions on the inhomogeneous term $ V $ with a small $ \epsilon > 0. $ We also demonstrate that these localized standing-waves converge to standing waves of the nonlinear Schrödinger equation with the homogeneous nonlinearity.

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Mathematics Subject Classification: Primary: 35B35, 35B60, 35Q40, 35Q55, 76B25.

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