  This paper was devoted to study the quantitative homogenization problems for
nonlinear elliptic operators in perforated domains. We obtained a sharp error
estimate $O(\varepsilon)$ when the problem was anchored in the reference domain
$\varepsilon\omega$. If concerning a bounded perforated domain, one will see a
bad influence from the boundary layers, which leads to the loss of the
convergence rate by $O(\varepsilon^{1/2})$. Equipped with the error estimates,
we developed both interior and boundary Lipschitz estimates at large-scales. As
an application, we received the so-called quenched Calder\'on-Zygumund
estimates by Shen's real arguments. To overcome some difficulties, we improved
the extension theory from (\cite[Theorem 4.3]{OSY}) to $L^p$-versions with
$\frac{2d}{d+1}-\epsilon<p<\frac{2d}{d-1}+\epsilon$ and $0<\epsilon\ll1$.
Appealing to this, we established Poincar\'e-Sobolev inequalities of local type
on perforated domains. Some of results in the present literature are new even
for related linear elliptic models.
