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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 $1<p<3$ withparameters $\lambda ,\mu_{ij}>0$. Novel approaches are employed to prove theexistence of a positive solution for $1<p<3$ including, particularly, thefinding of a ground state solution for $2\leq p<3$ using established linearalgebra techniques and demonstrating the existence of two distinct positivesolutions for $1<p<2.$ The analysis here, by employing alternative techniques,yields additional and improved results to those obtained in the study of Jinand Seok [Calc. Var. (2023) 62:72].

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