On the nonlinear Schr\"{o}dinger-Poisson systems with positron-electron interaction

TL;DR

This paper investigates the existence of positive solutions for a nonlinear Schrödinger-Poisson system with positron-electron interaction, introducing novel methods that improve upon previous results and find multiple solutions in certain parameter ranges.

Contribution

It develops new analytical techniques to establish the existence of positive solutions, including ground states and multiple solutions, for the nonlinear Schrödinger-Poisson system with specific nonlinearities.

Findings

Existence of a positive ground state solution for 2 ≤ p < 3.

Existence of two distinct positive solutions for 1 < p < 2.

Improved results over previous studies by Jin and Seok.

Abstract

We study the Schr\"{o}dinger-Poisson type system: \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+\lambda u+\left( \mu _{11}\phi _{u}-\mu _{12}\phi _{v}\right) u=% \frac{1}{2\pi }\int_{0}^{2\pi }\left\vert u+e^{i\theta }v\right\vert ^{p-1}\left( u+e^{i\theta }v\right) d\theta & \text{ in }\mathbb{R}^{3}, \\ -\Delta v+\lambda v+\left( \mu _{22}\phi _{v}-\mu _{12}\phi _{u}\right) v=% \frac{1}{2\pi }\int_{0}^{2\pi }\left\vert v+e^{i\theta }u\right\vert ^{p-1}\left( v+e^{i\theta }u\right) d\theta & \text{ in }\mathbb{R}^{3},% \end{array}% \right. \end{equation*}% where $1<p<3$ with parameters $\lambda ,\mu_{ij}>0$. Novel approaches are employed to prove the existence of a positive solution for $1<p<3$ including, particularly, the finding of a ground state solution for $2\leq p<3$ using established linear algebra techniques and demonstrating the existence of two distinct positive solutions for $1<p<2.$ The analysis here, by employing alternative techniques, yields additional and improved results to those obtained in the study of Jin and Seok [Calc. Var. (2023) 62:72].