Infinitely many solutions for Schr\"{o}dinger-Newton equations

TL;DR

This paper proves the existence of infinitely many non-radial positive solutions for the Schrödinger-Newton equations under specific conditions on the potential function at infinity, using a reduction method to construct multi-bump solutions.

Contribution

It establishes the existence of infinitely many non-radial solutions for Schrödinger-Newton equations with particular asymptotic potential behavior, introducing a novel reduction technique.

Findings

Existence of infinitely many non-radial solutions.

Construction of s-bump solutions on a circle of radius ~ (s log s)^{1/(1-m)}.

Solutions depend on the asymptotic behavior of the potential V(r).

Abstract

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