On the critical norm concentration for the inhomogeneous nonlinear Schrödinger equation

Abstract

We consider the inhomogeneous nonlinear Schrödiger equation (INLS) in ${\mathbb{R}}^N$ $$i\partial u_t+\Delta u+|x{|}^{-b}|u{|}^{2\sigma }u=0,$$ and show the $L^2$-norm concentration for the finite time blow-up solutions in the $L^2$-critical case, $\sigma =\frac{2-b}{N}$. Moreover, we provide an alternative for the classification of minimal mass blow-up solutions first proved by Genoud and Combet [4]. For the case $\frac{2-b}{N}<\sigma <\frac{2-b}{N-2}$, we show results regarding the $L^p$-critical norm concentration, generalizing the argument of Holmer and Roudenko [16] to the INLS setting.