  We consider the inhomogeneous nonlinear Schr\"odiger 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.
