  We investigate two equivalent capacitively and tunnel coupled quantum dots,
each coupled to its own pair of leads. Local Holstein type electron-phonon
coupling at the dots is assumed. To study many-body effects we use the finite-U
mean-field slave boson approach. For vanishing interdot interaction, weak e-ph
coupling and finite tunneling, molecular orbital spin Kondo effects occur for
single electron or single hole occupations. Phonons influence both correlations
and tunneling and additionally they shift the energies of the dots. Depending
on the dot energies and the strength of electron-phonon coupling, the system is
occupied by a different number of electrons that effectively interact with each
other repulsively or attractively leading to a number of different ground
states of DQD. Among them are Kondo-like states with spin, orbital or charge
correlations resulting from polaron cotunneling processes and states with
magnetic intersite correlations.
