# Infinitely many solutions for Schrödinger-Newton equations

created by jevnikar on 08 Jun 2021

[BibTeX]

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 $r\sim (s\log s)^{\frac{1}{1-m}}$.