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

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.