Mass-Critical Neutron Stars in the Hartree-Fock and Hartree-Fock-Bogoliubov Theories

We investigate the ground states of neutron stars and white dwarfs in the Hartree-Fock (HF) and Hartree-Fock-Bogoliubov (HFB) theories. It is known that the system is stable below a critical mass, which depends on the gravitational constant, while it becomes unstable if the total mass exceeds the critical mass. We prove that if the total mass is at the critical mass, then the HFB minimizers do not exist for any gravitational constant, while the HF minimizers exist for every gravitational constant except for a countable set, which is fully characterized by the Gagliardo-Nirenberg inequality for orthonormal systems. Our results complement the existence results in the sub-critical mass case established in [E. Lenzmann and M. Lewin, Duke Math. J., 2010].