  Using a variational Monte Carlo method, we study competitions of strong
electron-electron and electron-phonon interactions in the ground state of
Holstein-Hubbard model on a square lattice. At half filling, an extended
intermediate metallic or weakly superconducting (SC) phase emerges, sandwiched
by antiferromagnetic and charge order (CO) insulating phases. By the carrier
doping into the CO insulator, the SC order dramatically increases for strong
electron-phonon couplings, but largely hampered by wide phase separation (PS)
regions. Superconductivity is optimized at the border to the PS.
