We study the open-shell, spin-polarized Kohn-Sham models for non-relativistic and quasi-relativistic N-electron Coulomb systems, that is, systems where the kinetic energy of the electrons is given by either the non-relativistic operator −Δxn or the quasi-relativistic operator √−α−²Δxn + α−4 − α−². For standard and extended Kohn-Sham models in the local density approximation, we prove existence of a ground state (or minimizer) provided that the total charge Ztot of K nuclei is greater than N − 1. For the quasi-relativistic setting we also need that Ztot is smaller than a critical charge Zc = 2α−¹π−¹
