Solubility of non-linear elliptic systems in spaces that are weaker than the natural energy space

We introduce a scale of spaces that are dual to the classical Morrey space. We establish the solubility of non-linear elliptic systems on an interval of this scale, the range of the interval being essentially dependent on the modulus of ellipticity of the system. As a consequence, we prove solubility when the right-hand side is
(1) a Lebesgue space with exponent weaker than the Sobolev exponent,
(2) a space of densities of finite Borel measure, and
(3) a Hardy space for $p\leqslant 1$ under certain restrictions on the modulus of ellipticity.
We prove the existence and good behaviour of solutions of fundamental type. Our results are also completely new for linear systems with bounded discontinuous coefficients.