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Instability of standing waves of the Schrödinger equation with inhomogeneous nonlinearity

Authors:	Yue Liu Xiao-Ping Wang Ke Wang

Institution:	Department of Mathematics, University of Texas, Arlington, Texas 76019 ; Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong ; California Institute of Technology, MC 217-50, 1200 E. California Boulevard, Pasadena, California 91125

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 $\epsilon > 0.$

Keywords:	Nonlinear Schr\"odinger equation inhomogeneous nonlinearities blow-up standing waves ground state stability theory

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