Key words Proper nonlinear operators, quasilinear systems, compactness, decomposition lemma MSC (2010) 35G30, 35J55, 35A35The purpose of this paper is to provide tools for analyzing the compactness properties 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.

We consider a non-convex variational problem (P) and the corresponding singular perturbed problem (P ε ). The qualitative behavior of stable critical points of (P ε ) depending on ε and a lower order term is discussed and we prove compactness of a sequence of stable critical points as ε 0. Moreover we show whether this limit is the global minimizer of (P). Furthermore uniform convergence is considered as well as the convergence rate depending on ε.

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