Local Density Approximations of 3D Self-Gravitating Bose–Einstein Condensates

In the infinite three-dimensional space, we study a homogeneous nonrelativistic Schrödinger Hamiltonian describing a self-gravitating Bose gas with two types of interactions: a repulsive scaled interaction modeled by [Formula: see text] and an attractive Newtonian potential of the form [Formula: see text]. We investigate the mean-field limit as the number of particles [Formula: see text], with coupling parameters [Formula: see text] and [Formula: see text] satisfying [Formula: see text]. In this regime, we rigorously derive the corresponding Gross–Pitaevskii–Poisson theory. Furthermore, we analyze the asymptotic behavior of the condensate profile in two opposite limiting cases: when [Formula: see text], the system is well approximated by a Thomas–Fermi-type model, whereas when [Formula: see text], the condensate approaches a Choquard–Pekar-type profile.