Infinitely many solutions for Schrödinger-Newton equations

Abstract

We prove the existence of infinitely many non-radial positive solutions for the Schrödinger-Newton system $$ \left\{\begin{array}{ll} Δu- V(|x|)u + Ψu=0, &x\in\mathbb{R}^3,\newline ΔΨ+\frac12 u^2=0, &x\in\mathbb{R}^3, \end{array}\right. $$ provided that $V(r)$ has the following behavior at infinity: $$ V(r)=V_0+\frac{a}{r^m}+O\left(\frac{1}{r^{m+θ}}\right) \quad\mbox{ as } r\rightarrow\infty, $$ where $\frac12\le m<1$ and $a, V_0, θ$ 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 $r\sim (s\log s)^{\frac{1}{1-m}}$.