Existence of unstable standing waves for the inhomogeneous nonlinear Schrödinger equation

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January 2008, 7(1): 193-209. doi: 10.3934/cpaa.2008.7.193

Existence of unstable standing waves for the inhomogeneous nonlinear Schrödinger equation

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

Received January 2007 Revised June 2007 Published October 2007

Abstract
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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.

Keywords: Nonlienar Schrödinger equation, inhomogeneous nonlinearities, standing waves, stability..

Mathematics Subject Classification: Primary: 35B35, 35B60, 35Q40, 35Q55, 76B2.

Citation: Yue Liu. Existence of unstable standing waves for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (1) : 193-209. doi: 10.3934/cpaa.2008.7.193

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