Complex-valued Solutions of the Planar Schr\"odinger-Newton System

In this paper, we consider complex-valued solutions of the planar Schr\"odinger-Newton system, which can be described by minimizers of the constraint minimization problem. It is shown that there exists a critical rotational velocity $0<\Omega^*\leq \infty$, depending on the general trapping potential $V(x)$, such that for any rotational velocity $0\leq\Omega<\Omega^*$, minimizers exist if and only if $0<a<a^*:=\|Q\|_{2}^{2}$, where $Q>0$ is the unique positive solution of $-\Delta u+u-u^3=0$ in $\mathbb{R}^2$. Moreover, under some suitable assumptions on $V(x)$, applying blow-up analysis and energy estimates, we present a detailed analysis on the concentration behavior of minimizers as $a\nearrow a^*$.