  We study the effects of Kohn anomalies on the superconducting properties in
electron- and hole-doped cases of monolayer blue phosphorene, considering both
adiabatic and non-adiabatic phonon dispersions using first-principles
calculations. We show that the topology of the Fermi surface is crucial for the
formation of Kohn anomalies of doped blue phosphorene. By using the anisotropic
Eliashberg formalism, we further carefully consider the temperature dependence
of the non-adiabatic phonon dispersions. In cases of low hole densities, strong
electron-phonon coupling leads to a maximum critical temperature of $T_c=97$ K
for superconductivity. In electron-doped regimes, on the other hand, a maximum
superconducting critical temperature of $T_c=38$ K is reached at a doping level
that includes a Lifshitz transition point. Furthermore, our results indicate
that the most prominent component of electron-phonon coupling arises from the
coupling between an in-plane (out-of-plane) deformation and in-plane
(out-of-plane) electronic states of the electron (hole) type doping.
