Existence and Uniqueness of Constraint Minimizers for the Planar Schrodinger-Poisson System with Logarithmic Potentials

TL;DR

This paper investigates the existence and uniqueness of constraint minimizers for a planar Schrödinger-Poisson system with logarithmic potentials, identifying a critical threshold for the $L^2$ norm and analyzing local uniqueness near this threshold.

Contribution

It establishes the existence threshold for constraint minimizers in the logarithmic Schrödinger-Poisson system and analyzes their local uniqueness as the norm approaches this threshold.

Findings

Existence of a critical threshold $ ho^*$ for minimizers.

Constraint minimizers exist if and only if $0< ho< ho^*$.

Local uniqueness of positive minimizers near $ ho^*$.

Abstract

In this paper, we study constraint minimizers $u$ of the planar Schr\"odinger-Poisson system with a logarithmic convolution potential $\ln |x|\ast u^2$ and a logarithmic external potential $V(x)=\ln (1+|x|^2)$, which can be described by the $L^2$-critical constraint minimization problem with a subcritical perturbation. We prove that there is a threshold $\rho ^* \in (0,\infty)$ such that constraint minimizers exist if and only if $0<\rho<\rho^*$. In particular, the local uniqueness of positive constraint minimizers as $\rho\nearrow\rho^*$ is analyzed by overcoming the sign-changing property of the logarithmic convolution potential and the non-invariance under translations of the logarithmic external potential.