Stationary solutions of the Schrödinger-Newton model – an ODE approach. (English) Zbl 1224.35385

Choquard, Philippe; Stubbe, Joachim; Vuffray, Marc

Differ. Integral Equ. 21, No. 7-8, 665-679 (2008).

Summary: 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 \(d=3\).

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35Q40	PDEs in connection with quantum mechanics
47J10	Nonlinear spectral theory, nonlinear eigenvalue problems

Keywords:

Schrödinger-Newton model; spherically symmetric solution; existence; uniqueness