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We consider the following time-independent nonlinear $L^2$-criticalSchr\"{o}dinger equation \[ -\Deltau(x)+V(x)u(x)-a|x|^{-b}|u|^{1+\frac{4-2b}{N}}=\mu u(x)\,\ \hbox{in}\,\\mathbb{R}^N, \] 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 theabove equation can be equivalently described by minimizers of the correspondingminimization 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 dependssensitively on the value of $V(0)$. Moreover, when there is no minimizer atthreshold $a^*$, we give a detailed concentration behavior of minimizers as$a\nearrow a^*$, based on which we finally prove that there is a uniqueminimizer as $a\nearrow a^*$.

Concentration Behavior of Ground States for $L^2$-Critical Schr\"{o}dinger Equation with a Spatially Decaying Nonlinearity