  A microscopic theory for the coupling of intrinsic Josephson oscillations and
dispersive phonon branches in layered superconductors is developed. Thereby the
effect of phonons on the electronic c-axis transport enters through an
effective longitudinal dielectric function. This coupling provides an
explanation of recently observed subgap resonances in the $I_{dc}$-$V_{dc}$-
curve of anisotropic cuprate superconductors forming a stack of short Josephson
junctions. Due to the finite dispersion these resonances can appear at
van-Hove-singularities of both optical and acoustical phonon branches,
explaining low-voltage structures in the I-V-characteristic, which are not
understood in phonon models without dispersion. In long junctions the
dispersion of collective electron-phonon modes parallel to the layers is
investigated.
