In this paper, we consider complex-valued solutions of the planar Schrödinger–Newton system, which can be described by minimizers of the constraint minimization problem. It is shown that there exists a critical rotational velocity \(0<\Omega ^*\le \infty \) , depending on the general trapping potential V(x), such that for any rotational velocity \(0\le \Omega <\Omega ^*\) , minimizers exist if and only if \(0<a<a^*:=\Vert Q\Vert _{2}^{2}\) , 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 \(a\nearrow a^*\) .
