  We focus on optimal control problems governed by elliptic, quasilinear PDEs.
Though there are various examples of such problems in the literature, we make
an attempt at describing some general principles by dealing with three basic
situations. In the first one, we assume that the state equation is variational;
the second one focuses on a non-variational, monotone operator as state
equation; finally, we add a non-linear term off the divergence part of the
equation. In the first two cases, existence of optimal solutions can be
established under suitable sets of assumptions, while relaxation is required
for the third situation. Concerning the cost functional, and though more
general examples can be dealt with, we will take a typical case consisting of
two terms: one depending on the state, and another one of the form of a typical
Thychonov regularization.
