Differential and Integral Equations

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

Philippe Choquard, Joachim Stubbe, and Marc Vuffray

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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

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

First available in Project Euclid: 20 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
Secondary: 34A34: Nonlinear equations and systems, general 34L40: Particular operators (Dirac, one-dimensional Schrödinger, etc.) 47J20: Variational and other types of inequalities involving nonlinear operators (general) [See also 49J40]


Choquard, Philippe; Stubbe, Joachim; Vuffray, Marc. Stationary solutions of the Schrödinger-Newton model---an ODE approach. Differential Integral Equations 21 (2008), no. 7-8, 665--679. https://projecteuclid.org/euclid.die/1356038617

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