  An analytical method of studying strong long-range electron-phonon and
Coulomb interactions in complex lattices is presented. The method is applied to
a perovskite layer with anisotropic coupling of holes to the vibrations of
apical atoms. Depending on the relative strength of the polaronic shift, Ep,
and the inter-site Coulomb repulsion, Vc, the system is either a polaronic
Fermi liquid, Vc > 1.23 Ep, a bipolaronic superconductor, 1.16 Ep < Vc < 1.23
Ep, or a charge segregated insulator, Vc < 1.16 Ep. In the superconducting
window, the carriers are mobile bipolarons with a remarkably low effective
mass. The model describes the key features of the underdoped superconducting
cuprates.
