On the critical norm concentration for the inhomogeneous nonlinear Schr\"odinger equation

We consider the inhomogeneous nonlinear Schr"odiger equation (INLS) in ${\mathbb{R}}^N$ $$i\mathrm{\partial }{u}_t+\mathrm{\Delta }u+\mathrm{\mid }x{\mathrm{\mid }}^{-b}\mathrm{\mid }u{\mathrm{\mid }}^{2\sigma }u=0,$$ 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 $\frac{2-b}{N} < \sigma < \frac{2-b}{N-2}$,weshowresultsregardingthe$L^p$-critical norm concentration, generalizing the argument of Holmer and Roudenko [16] to the INLS setting.