  We study the ground state solutions of the Dirac-Fock model in the case of
weak electronic repulsion, using bifurcation theory. They are solutions of a
min-max problem. Then we investigate a max-min problem coming from the
electron-positron field theory of Bach-Barbaroux-Helffer-Siedentop. We show
that given a radially symmetric nuclear charge, the ground state of Dirac-Fock
solves this max-min problem for certain numbers of electrons. But we also
exhibit a situation in which the max-min level does not correspond to a
solution of the Dirac-Fock equations together with its associated
self-consistent projector.
