  The purpose of this paper is to provide tools for analyzing the compactness
of sequences in Sobolev spaces, in particular if the sequence gets mapped onto
a compact set by some nonlinear operator. Here, our focus lies on a very
general class of nonlinear operators arising in quasilinear systems of partial
differential equations of second order, in divergence form. Our approach, based
on a suitable decomposition lemma, admits the discussion of problems with some
inherent loss of compactness, for example due to a domain with infinite measure
or a lower order term with critical growth. As an application, we obtain a
characterization of properness which is considerably easier to verify than the
definition. The methods presented can also be used to check Palais--Smale
conditions for variational problems.
