Nonlinear Schr�dinger equations and sharp interpolation estimates

Abstract

A sharp sufficient condition for global existence is obtained for the nonlinear Schrodinger 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.