  We introduce the N-body simulation technique to follow structure formation in
linear and nonlinear regimes for the extended quintessence models
(scalar-tensor theories in which the scalar field has a self-interaction
potential and behaves as dark energy), and apply it to a class of models
specified by an inverse power-law potential and a non-minimal coupling. Our
full solution of the scalar field perturbation confirms that, when the
potential is not too nonlinear, the effects of the scalar field could be
accurately approximated as a modification of background expansion rate plus a
rescaling of the effective gravitational constant relevant for structure
growth. For the models we consider, these have opposite effects, leading to a
weak net effect in the linear perturbation regime. However, on the nonlinear
scales the modified expansion rate dominates and could produce interesting
signatures in the matter power spectrum and mass function, which might be used
to improve the constraints on the models from cosmological data. We show that
the density profiles of the dark matter halos are well described by the
Navarro-Frenk-White formula, although the scalar field could change the
concentration. We also derive an analytic formula for the scalar field
perturbation inside halos assuming NFW density profile and sphericity, which
agrees well with numerical results if the parameter is appropriately tuned. The
results suggest that for the models considered, the spatial variation of the
scalar field (and thus the locally measured gravitational constant) is very
weak, and so local experiments could see the background variation of
gravitational constant.
