  Two-dimensional (2D) metals can host gapless plasmonic excitations, which
strongly couple to electrons and thus may significantly affect
superconductivity in layered materials. To investigate the dynamical interplay
of the electron-electron and electron-phonon interactions in the theory of 2D
superconductivity, we apply a full momentum- and frequency-dependent one-loop
theory treating electron-phonon, electron-plasmon, and phonon-plasmon coupling
with the same accuracy. We tune the strength of the Coulomb interaction by
varying the external screening $\varepsilon_{ext}$ to the layered
superconductor and find three distinct regions. At weak screening,
superconductivity is mediated by plasmons. In the opposite limit conventional
electron-phonon interactions dominate. In between, we find a suppressed
superconducting state. Our results show that even conventional electron-phonon
mediated layered superconductors can be significantly affected by the
electron-plasmon coupling in a weak screening environment. This unconventional
pairing contribution can then be controlled by the external screening.
