  We study structure formation in non-minimally coupled dark energy models,
where there is a coupling in the Lagrangian between a quintessence scalar field
and gravity via the Ricci scalar. We consider models with a range of different
non-minimal coupling strengths and compare these to minimally coupled
quintessence models with time-dependent dark energy densities. The equations of
state of the latter are tuned to either reproduce the equation of state of the
non-minimally coupled models or their background history. Thereby they provide
a reference to study the unique imprints of coupling on structure formation. We
show that the coupling between gravity and the scalar field, which effectively
results in a time-varying gravitational constant G, is not negligible and its
effect can be distinguished from a minimally coupled model. We extend previous
work on this subject by showing that major differences appear in the
determination of the mass function at high masses, where we observe differences
of the order of 40% at z=0. Our new results concern effects on the non-linear
matter power spectrum and on the lensing signal (differences of ~10% for both
quantities), where we find that non-minimally coupled models could be
distinguished from minimally coupled ones.
