Dual Morrey spaces in nonlinear elliptic PDEs

We will talk on the following results:
For nonlinear elliptic equations and systems with a Hilbert energy space, we establish the solvability in dual Morrey spaces on an interval of scale that explicitly depends on the modulus of ellipticity (in the case of second-order systems, the dependence is exact). This gives the existence of solutions for a wider class of right-hand sides than previously known, e.g. from the Lebesgue spaces with an exponent weaker than the Sobolev exponent, or from Hardy classes for a certain interval $p<1$.
For nonlinear elliptic equations and systems with non-Hilbert energy space ($W^m_p$, $p\ne 2$) we establish existence of very weak solutions in dual Morrey spaces.
For linear elliptic equations and systems with discontinuous coefficients, we establish the existence of higher-order derivatives of solution; the increase of order depends explicitly on the modulus of ellipticity. The coefficients are assumed to be in the dual Morrey spaces with Lebesgue exponent infinity. They can have e.g. a dense set of discontinuities of type $x/|x|$, as in the counterexamples of Souček and John-Malý-Stará.