One of the aims of density functional theory is to obtain properties of (ground) states of large systems, in particular their energy, by solving a nonlinear equation, involving only the parameters of a single electron. The oldest such theory is the Thomas-Fermi approach. Major developments are due to Hohenberg, Kohn and Sham in the mid-1960s and the present article discusses an approximate nonlinear equation arising within the context of the Kohn-Sham approach [P. Hohenberg and W. Kohn, Phys. Rev. (2) 136 (1964), B864-B871; MR0180312 (31 #4547); W. Kohn and L. J. Sham, Phys. Rev. (2) 140 (1965), A1133-A1138; MR0189732 (32 #7154)]. It is connected to the exchange contribution to the Kohn-Sham functional. The latter arises from taking into account the Fermi-Dirac statistics and complicates the application of a variational procedure significantly. Thus it is often replaced by an approximate term and the authors discuss the existence of a solution for this situation. They make a few physically reasonable assumptions and are then able to demonstrate the existence of a proper solution.