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

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 \(d=3\).


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