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 the one available
for linear elliptic problems with continuous coefficients, 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.