  In nuclear physics, the relativistic mean-field theory describes the nucleus
as a system of Dirac nucleons which interact via meson fields. In a static case
and without nonlinear self-coupling of the $\sigma$ meson, the relativistic
mean-field equations become a system of Dirac equations where the potential is
given by the meson and photon fields. The aim of this work is to prove the
existence of solutions of these equations. We consider a minimization problem
with constraints that involve negative spectral projectors and we apply the
concentration-compactness lemma to find a minimizer of this problem. We show
that this minimizer is a solution of the relativistic mean-field equations
considered.
