• Proper nonlinear operators;
  • quasilinear systems;
  • compactness;
  • decomposition lemma;
  • MSC (2010) 35G30;
  • 35J55;
  • 35A35


The 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. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim