On properness and related properties of quasilinear systems on unbounded domains

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.