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

Abstract

We consider ground states of pseudo-relativistic boson stars with a self-interacting potential $K(x)$ in ${\mathbb{R}}^3$, 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 $0<N<N^{⁎}$, and there is no minimizer if $N>N^{⁎}$, where $N^{⁎}$ is called the critical stellar mass. In contrast to the case of the Coulomb-type potential where $K(x)\equiv 1$, 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.