On the nonlinear Schrödinger-Poisson systems with positron-electron interaction

Abstract

We study the Schrödinger-Poisson type system: \begin{equation*} \left\{ \begin{array}{ll} -Δu+λu+\left( μ_{11}ϕ_{u}-μ_{12}ϕ_{v}\right) u=% \frac{1}{2π}\int_{0}^{2π}\left\vert u+e^{iθ}v\right\vert ^{p-1}\left( u+e^{iθ}v\right) dθ& \text{ in }\mathbb{R}^{3}, \\ -Δv+λv+\left( μ_{22}ϕ_{v}-μ_{12}ϕ_{u}\right) v=% \frac{1}{2π}\int_{0}^{2π}\left\vert v+e^{iθ}u\right\vert ^{p-1}\left( v+e^{iθ}u\right) dθ& \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 $2\le p<3$ 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].