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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 $$d=3$$.

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