  We consider the case of a density-driven metal-superconductor transition in
the proximity of an electronic phase separation. In particular we investigate
the interplay between superconducting fluctuations and density fluctuations,
which become quantum critical when the electronic phase separation vanishes at
zero temperature into a quantum critical point. In this situation the critical
dynamical density fluctuations strongly affect the dynamics of the Cooper pair
fluctuations, which acquire a more singular character with a z=3 dynamical
critical index. This gives rise to a scenario that possibly rules the
disappearance of superconductivity when the electron density is reduced by
elecrostatic gating at the LaAlO3/SrTiO3 interface.
