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−2Deltaxn+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 Zmtot of K nuclei is greater than N-1 and that Zmtot is smaller than a critical charge Zmc=2alpha−1pi−1