Nonlinear ellitpic equations beyond the natural duality pairing

Many real-world problems are described by nonlinear partial differential equations. A promiment example of such equations is nonlinear (quasilinear) elliptic system with given right hand side in divergence form div f data. In case data are good enough (i.e., belong to L^2), one can solve such a problem by using the monotone operator therory, however in case data are worse no existence theory was available except the case when the operator is linear, e.g. the Laplace operator. For this particular case one can however establish the existence of a solution whose gradient belongs to L^q whenever f belongs to L^q as well. From this point of view it would be nice to have such a theory also for general operators. However, it cannot be the case as indicated by many counterexamples. Nevertheless, we show that such a theory can be built for operators having asymptotically the Uhlenbeck structrure, which is a natural class of operators in the theory of PDE. As a by product we develop new theoretical tools as e.g., weighted estimates for the linear problems and the new compensated compactness method represented by the div-curl-biting-weighted lemma.