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Nonlinear Schrödinger equations and sharp interpolation estimates

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Abstract

A sharp sufficient condition for global existence is obtained for the nonlinear Schrödinger equation

$$\begin{array}{*{20}c} {(NLS)} & {2i\phi _t + \Delta \phi + \left| \phi \right|^{2\sigma } \phi = 0,} & {x \in \mathbb{R}^N } & {t \in \mathbb{R}^ + } \\ \end{array} $$

in the case σ=2/N. This condition is in terms of an exact stationary solution (nonlinear ground state) of (NLS). It is derived by solving a variational problem to obtain the “best constant” for classical interpolation estimates of Nirenberg and Gagliardo.

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  1. Michael I. Weinstein

    Present address: Department of Mathematics, Stanford University, 94305, Stanford, CA, USA

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  1. Courant Institute of Mathematical Sciences, New York University, 10012, New York, NY, USA

    Michael I. Weinstein

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  1. Michael I. Weinstein
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Communicated by J. Moser

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Weinstein, M.I. Nonlinear Schrödinger equations and sharp interpolation estimates. Commun.Math. Phys. 87, 567–576 (1983). https://doi.org/10.1007/BF01208265

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