  The thermodynamic impact of the Coulomb repulsion on s-wave superconductors
is analyzed via a rigorous study of equilibrium and ground states of the strong
coupling BCS-Hubbard Hamiltonian. We show that the one-site electron repulsion
can favor superconductivity at fixed chemical potential by increasing the
critical temperature and/or the Cooper pair condensate density. If the one-site
repulsion is not too large, a first or a second order superconducting phase
transition can appear at low temperatures. The Mei\ss ner effect is shown to be
rather generic but coexistence of superconducting and ferromagnetic phases is
also shown to be feasible, for instance near half-filling and at strong
repulsion. Our proof of a superconductor-Mott insulator phase transition
implies a rigorous explanation of the necessity of doping insulators to create
superconductors. These mathematical results are consequences of "quantum large
deviation" arguments combined with an adaptation of the proof of St{\o}rmer's
theorem to even states on the CAR algebra.
