  We consider the focusing inhomogeneous nonlinear Schr\"odinger (INLS)
equation in $\mathbb{R}^N$ $$i \partial_t u +\Delta u + |x|^{-b} |u|^{2\sigma}u
= 0,$$ where $N\geq 2$ and $\sigma$, $b>0$. We first obtain a small data global
result in $H^1$, which, in the two spatial dimensional case, improves the third
author result in [22] on the range of $b$. For $N\geq 3$ and
$\frac{2-b}{N}<\sigma<\frac{2-b}{N-2}$, we also study the local well posedness
in $\dot H^{s_c}\cap \dot H^1 $, where $s_c=\frac{N}{2}-\frac{2-b}{2\sigma}$.
Sufficient conditions for global existence of solutions in $\dot H^{s_c}\cap
\dot H^1$ are also established, using a Gagliardo-Nirenberg type estimate.
Finally, we study the $L^{\sigma_c}-$norm concentration phenomenon, where
$\sigma_c=\frac{2N\sigma}{2-b}$, for finite time blow-up solutions in $\dot
H^{s_c}\cap \dot H^1$ with bounded $\dot H^{s_c}-$norm. Our approach is based
on the compact embedding of $\dot H^{s_c}\cap \dot H^1$ into a weighted
$L^{2\sigma+2}$ space.
