Is evolution from weak to strong coupling superconductivity always continuous?

We have performed a variational analysis on the evolution of superconductivity from weak to strong coupling regime. We have improved the Bardeen Cooper Schrieffer (BCS) wave function, and show that the electronic correlations are important in obtaining the correct behavior of superconducting condensation energy. In contrast to a crossover without thennodynamic anomaly discussed in a dilute system, we show the existence of a quantum phase transition near half filling, which suggests that an evolution is not always continuous. The transition is driven by charge density waves instabilities. We have found that superconductivity and charge density waves coexist in the presence of a weak intersite repulsion. The ground state phase diagram is determined by varying both interact ions and filling.