Normalized solutions for the planar Schrödinger-Poisson system with two electrons interaction
Baihong Li, Yuanhong Wei, Xiangjian Zeng
TL;DR
This work studies normalized (mass-constrained) solutions to the planar Schrödinger-Poisson system with two-electron interaction in $ R^2$, where a logarithmic convolution couples the components. The authors develop a tailored variational framework on $E imes E$ to handle the nonlocal log kernel, prove compactness and the existence of a global minimizer (ground state) under broad parameter regimes, and establish a mass-supercritical multiplicity result. A new Pohozaev identity for the coupled system with the logarithmic term underpins the energy estimates and the minimax construction. Together, these results extend normalized-solution theory to a two-component planar SP system with a nonlocal logarithmic interaction and demonstrate dual existence (ground and excited states) in the mass-supercritical regime.
Abstract
This paper focuses on the normalized solutions for the planar Schrödinger-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 Pohožaev identity for the coupled Schrödinger-Poisson system with a logarithmic convolution term is also shown, which is crucial for addressing the mass-supercritical problem.