  The strong coupling Eliashberg theory plus vertex correction is used to
calculate maps of transition temperature (Tc) in parameter-space characterizing
superconductivity. Based on these Tc maps, crossover behaviors are found when
electron-phonon interaction increases from weak-coupling region to strong
coupling region. Especially, the combined interaction of vertex correction and
Coulomb interaction can efficiently depress Tc from extremely high values in
standard strong-coupling theory to reasonable values found in experiments and
successfully explain the doing-dependent Tc of cuprate superconductors. The
strong non-adiabatic effect is the barrier for high-Tc in compounds with
compositions of light atoms and with high phonon frequencies.
