We undertake the mathematical analysis of a model describing equilibrium binary electrolytes surrounded by charged solid walls. The problem is formulated in terms of the electrostatic potential and the ionic concentrations which have prescribed spatial mean values. The free energy of the system is decomposed as the difference of the internal energy and entropy functionals. The entropy functional is the sum of an ideal entropy and an excess entropy, the latter taking into account non-ideality due to electrostatic correlations at low ionic concentrations and steric exclusion effects at high ionic concentrations. We derive sufficient conditions to achieve convexity of the entropy functional, yielding a convex-concave free energy functional. Our main result is the existence and uniqueness of the saddle point of the free energy functional and its characterization as a solution of the original model problem. The proof hinges on positive uniform lower bounds for the ionic concentrations and uniform upper bounds for the ionic concentrations and the electrostatic potential. Some numerical experiments are presented in the case where the excess entropy is evaluated using the Mean Spherical Approximation.