Normalized solutions for the planar Schr\"{o}dinger-Poisson system with two electrons interaction

TL;DR

This paper investigates the existence of normalized solutions for the planar Schrödinger-Poisson system with two-electron interactions, establishing ground and excited state solutions through advanced compactness and Pohozaev identity methods.

Contribution

It introduces new existence results for solutions in the Schrödinger-Poisson system with logarithmic convolution, including the development of a compactness method and Pohozaev identity for mass-supercritical cases.

Findings

Existence of a ground state solution for general cases.

Existence of two solutions in the mass-supercritical case, including an excited state.

Development of a compactness method for logarithmic convolution functionals.

Abstract

This paper focuses on the normalized solutions for the planar Schr\"{o}dinger-Poisson system with a two-electron interaction, which models the effect between electrons and the electrostatic potential they generate. As the parameters vary, some existence results are established. Specifically, a ground state solution is obtained for some general cases. The existence of two solutions is established for the mass-supercritical case, one of which is a ground state solution and the other one is an excited state solution. We develop a compactness method to deal with the functionals involving logarithmic convolution terms. The Poho\v{z}aev identity for the coupled Schr\"{o}dinger-Poisson system with a logarithmic convolution term is also shown, which is crucial for addressing the mass-supercritical problem.