*Preprint*

**Inserted:** 8 jun 2021

**Last Updated:** 8 jun 2021

**Year:** 2021

**Abstract:**

We prove the existence of infinitely many non-radial positive solutions for the Schrödinger-Newton system

$ \begin{cases}
\Delta u- V(

x

)u + \Psi u=0,\quad &x\in\mathbb{R}^3,\\
\Delta \Psi+\frac12 u^2=0, &x\in\mathbb{R}^3,
\end{cases}
$

provided that $V(r)$ has the following behavior at infinity:

$ V(r)=V_0+\frac{a}{r^m}+O\left(\frac{1}{r^{m+\theta}}\right) \quad\mbox{ as } r\rightarrow\infty, $

where $\frac12\le m<1$ and $a, V_0, \theta$ are some positive constants. In particular, for any $s$ large we use a reduction method to construct $s-$bump solutions lying on a circle of radius $r\sim (s\log s)^{\frac{1}{1-m}}$.

**Keywords:**
infinitely many solutions, reduction method, perturbation problem, Schrödinger-Newton system

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