## Differential and Integral Equations

### Stationary solutions of the Schrödinger-Newton model---an ODE approach

#### Abstract

We prove the existence and uniqueness of stationary spherically symmetric positive solutions for the Schrödinger-Newton model in any space dimension $d$. Our result is based on an analysis of the corresponding system of second-order differential equations. It turns out that $d=6$ is critical for the existence of finite energy solutions and the equations for positive spherically symmetric solutions reduce to a Lane-Emden equation for all $d\geq 6$. Our result implies, in particular, the existence of stationary solutions for two-dimensional self-gravitating particles and closes the gap between the variational proofs in $d=1$ and $d=3$.

#### Article information

Source
Differential Integral Equations, Volume 21, Number 7-8 (2008), 665-679.

Dates
First available in Project Euclid: 20 December 2012