Existence of a minimizer for the quasi-relativistic Kohn-Sham model

We study the standard and extended Kohn-Sham models for quasi-relativistic N-electron Coulomb systems; that is, systems where the kinetic energy of the electrons is given by the quasi-relativistic operator $$ sqrt{-alpha^{-2}Delta_{x_n}+alpha^{-4}}-alpha^{-2}. $$ For spin-unpolarized systems in the local density approximation, we prove existence of a ground state (or minimizer) provided that the total charge $Z_{ m tot}$ of K nuclei is greater than N-1 and that $Z_{ m tot}$ is smaller than a critical charge $Z_{ m c}=2 alpha^{-1} pi^{-1}$.