Calculus of Variations and Geometric Measure Theory

Y. Hu - A. Jevnikar - W. Xie

Infinitely many solutions for Schrödinger-Newton equations

created by jevnikar on 08 Jun 2021
modified on 01 Apr 2022


Accepted Paper

Inserted: 8 jun 2021
Last Updated: 1 apr 2022

Journal: Commun. Contemp. Math.
Year: 2021


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

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