Ground states of pseudo-relativistic boson stars under the critical stellar mass

We consider ground states of pseudo-relativistic boson stars with a self-interacting potential K(x) in R3, which can be described by minimizers of the pseudo-relativistic Hartree energy functional. Under some assumptions on K(x), minimizers exist if the stellar mass N satisfies 0N⁎, where N⁎ is called the critical stellar mass. In contrast to the case of the Coulomb-type potential where K(x)≡1, we prove that the existence of minimizers may occur at N=N⁎, depending on the local profile of K(x) near the origin. When there is no minimizer at N=N⁎, we also present a detailed analysis of the behavior of minimizers as N approaches N⁎ from below, for which the stellar mass concentrates at a unique point.