On Kohn-Sham models with LDA and GGA exchange-correlation functionals

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.