  A variational approach is proposed to determine some properties of the
adiabatic Holstein-Hubbard model which describes the interactions between a
static atomic lattice and an assembly of fermionic charge carriers. The sum of
the electronic energy and the lattice elastic energy is proved to have minima
with a polaron structure in a certain domain of the phase diagram. Our
analytical work consists in the expansion of these energy minima from the zero
electronic transfer limit which remarkably holds for a finite amplitude of the
onsite Hubbard repulsion and for an unbounded lattice size.
