  We establish existence, uniqueness and optimal regularity results for very
weak solutions to certain nonlinear elliptic boundary value problems. We
introduce structural asymptotic assumptions of Uhlenbeck type on the
nonlinearity, which are sufficient and in many cases also necessary for
building such a theory. We provide a unified approach that leads qualitatively
to the same theory as that one available for linear elliptic problems with
continuous coeffcients, e.g. the Poisson equation. The result is based on
several novel tools that are of independent interest: local and global
estimates for (non)linear elliptic systems in weighted Lebesgue spaces with
Muckenhoupt weights, a generalization of the celebrated div{curl lemma for
identification of a weak limit in border line spaces and the introduction of a
Lipschitz approximation that is stable in weighted Sobolev spaces.
