On ground states for the L²-critical boson star equation

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We consider ground state solutions u ≥ 0 for the L²-critical boson star equation √-Δ u - (|x|⁻¹ |u|²) u = -u in ³. We prove analyticity and radial symmetry of u. In a previous version of this paper, we also stated uniqueness and nondegeneracy of ground states for the L²-critical boson star equation in ³, but the arguments given there contained a gap. However, we refer to our recent preprint FraLe in arXiv:1009.4042, where we prove a general uniqueness and nondegeneracy result for ground states of nonlinear equations with fractional Laplacians in d=1 space dimension.