Complex-valued Solutions of the Planar Schrödinger-Newton System

Abstract

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 $0<{\Omega }^{\ast }\le \infty$, depending on the general trapping potential $V(x)$, such that for any rotational velocity $0\le \Omega <{\Omega }^{\ast }$, minimizers exist if and only if $0<a<a^{\ast }:=\Vert Q{\Vert }_2^2$, 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 $a\nearrow a^{\ast }$.

arXiv admin note: text overlap with arXiv:2212.00234 by other authors