On the Critical Norm Concentration for the Inhomogeneous Nonlinear Schrödinger Equation

Abstract

We consider the inhomogeneous nonlinear Schrödinger equation (INLS) in \({\mathbb {R}}^N\)$$\begin{aligned} i \partial _t u + \Delta u + |x|^{-b} |u|^{2\sigma }u = 0, \end{aligned}$$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 proof for the classification of minimal mass blow-up solutions, first proved by Genoud and Combet (J Evol Equ 16(2):483–500, 2016, https://doi.org/10.1007/s00028-015-0309-z). 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 (Appl Math Res eXpress 2007(1):Art. ID abm004, 2007) to the INLS setting.