  We examine the superconducting solution in the Holstein model, where the
conduction electrons couple to the dispersionless Boson fields, using the
Migdal-Eliashberg theory and Dynamical Mean Field Theory. Although different in
numerical values, both methods imply the existence of an optimal Boson energy
for superconductivity at a given electron-Boson coupling. This non-monotonous
behavior can be understood as an interplay between the polaron and
superconducting physics, as the electron-Boson coupling is the origin of the
superconductor, but at the same time traps the conduction electrons making the
system more insulating. Our calculation provides a simple explanation on the
recent experiment on sulfur hydride (H$_2$S), where an optimal pressure for the
superconductivity was observed. The validities of both methods are discussed.
