  This article is concerned with the mathematical analysis of the Kohn-Sham and
extended Kohn-Sham models, in the local density approximation (LDA) and
generalized gradient approximation (GGA) frameworks. After recalling the
mathematical derivation of the Kohn-Sham and extended Kohn-Sham LDA and GGA
models from the Schr\"odinger equation, we prove that the extended Kohn-Sham
LDA model has a solution for neutral and positively charged systems. We then
prove a similar result for the spin-unpolarized Kohn-Sham GGA model for
two-electron systems, by means of a concentration-compactness argument.
