Concentration Behavior of Ground States for $L^2$-Critical Schrödinger Equation with a Spatially Decaying Nonlinearity

Abstract: 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 $a\nearrow a^*$.