  We consider the following time-independent nonlinear $L^2$-critical
Schr\"{o}dinger equation \[ -\Delta
u(x)+V(x)u(x)-a|x|^{-b}|u|^{1+\frac{4-2b}{N}}=\mu u(x)\,\ \hbox{in}\,\
\mathbb{R}^N, \] where $\mu\in\mathbb{R}$, $a>0$, $N\geq 1$, $0<b<\min\{2,N\}$,
and $V(x)$ is an external potential. It is shown that ground states of the
above equation can be equivalently described by minimizers of the corresponding
minimization problem. In this paper, we prove that there is a threshold $a^*>0$
such that minimizer exists for $0<a<a^*$ and minimizer does not exist for any
$a>a^*$. However if $a=a^*$, it is proved that whether minimizer exists depends
sensitively on the value of $V(0)$. Moreover, when there is no minimizer at
threshold $a^*$, we give a detailed concentration behavior of minimizers as
$a\nearrow a^*$, based on which we finally prove that there is a unique
minimizer as $a\nearrow a^*$.
