  Aims. Exoplanet-host stars (EHS) are known to present superficial chemical
abundances different from those of stars without any detected planet (NEHS).
EHS are, on the average, overmetallic compared to the Sun. The observations
also show that, for cool stars, lithium is more depleted in EHS than in NEHS.
The aim of this paper is to obtain constraints on possible models able to
explain this difference, in the framework of overmetallic models compared to
models with solar abundances. Methods. We have computed main sequence stellar
models with various masses and metallicities. The results show different
behaviour for the lithium destruction according to those parameters. We compare
these results to the spectroscopic observations of lithium. Results. Our models
show that the observed lithium differences between EHS and NEHS are not
directly due to the overmetallicity of the EHS: some extra mixing is needed
below the convective zones. We discuss possible explanations for the needed
extra mixing, in particular an increase of the mixing efficiency associated
with the development of shear instabilities below the convective zone,
triggered by angular momentum transfer due to the planetary migration.
