Existence of infinitely many distinct solutions to the quasi-relativistic Hartree-Fock equations

We establish existence of infinitely many distinct solutions to the Hartree-Fock equations for Coulomb systems with quasi-relativistic kinetic energy $\sqrt{ -\a^{-2} D_{x_{n}} + \a^{-4}} -\a^{-2}$ for the $n^{\rm th}$ electron. Moreover, we prove existence of a ground state. The results are valid under the hypotheses that the total charge $Z_{\rm tot}$ of $K$ nuclei is greater than or equal to the total number of electrons $N$ and that $Z_{\rm tot}$ is smaller than some critical charge $Z_{\rm c}$. The proofs are based on critical point theory in combination with density operator techniques.