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New type of solutions for the nonlinear Schrödinger-Newton system

The nonlinear Schrödinger-Newton system \begin{equation*} \begin{cases} Δu- V(|x|)u + Ψu=0, &~x\in\mathbb{R}^3,\\ ΔΨ+\frac12 u^2=0, &~x\in\mathbb{R}^3, \end{cases} \end{equation*} is a nonlinear system obtained by coupling the linear Schrödinger equation of quantum mechanics with the gravitation law of Newtonian mechanics. Wei and Yan in (Calc. Var. Partial Differential Equations 37 (2010),423--439) proved that the Schrödinger equation has infinitely many positive solutions in $\mathbb{R}^N$ and these solutions have polygonal symmetry in the $(y_{1}, y_{2})$ plane and they are radially symmetric in the other variables. Duan et al. in (arXiv:2006.16125v1) extended the results got by Wei and Yan and these solutions have polygonal symmetry in the $(y_{1}, y_{2})$ plane and they are even in $y_{2}$with one more more parameter in the expression of the solutions.Hu et al. Under the appropriate assumption on the potential function V, Hu et al. in (arXiv: 2106.04288v1) constructed infinitely many non-radial positive solutions for the Schrödinger-Newton system and these positive solutions have polygonal symmetry in the $(y_{1}, y_{2})$ plane and they are even in $y_{2}$ and $y_{3}$. Assuming that $V(r)$ has the following character \begin{equation*} V(r)=V_{1}+\frac{b}{r^q}+O\Big(\frac{1}{r^{q+σ}}\Big),~\mbox{ as } r\rightarrow\infty, \end{equation*} Where $\frac12\leq q<1$ and $b, V_{1}, σ$ are some positive constants, $V(y)\geq V_1>0$, we construct infinitely many non-radial positive solutions which have polygonal symmetry in the $(y_{1}, y_{2})$ plane and are even in $y_{2}$ for the Schrödinger-Newton system by the Lyapunov-Schmidt reduction method. We extend the results got by Duan et al. in (arXiv:2006.16125v1) to the nonlinear Schrödinger-Newton system.