Sobolev-Morrey regularity of solutions to general quasilinear elliptic equations

We deal with the Dirichlet problem for general quasilinear elliptic equations over Reifenberg flat domains. The principal part of the operator supports natural gradient growth and its x-discontinuity is of small-BMO type, while the lower order terms satisfy controlled growth conditions with x-behaviour modelled by Morrey spaces. We obtain a Calderòn–Zygmund type result for the gradient of the weak solution by proving that the solution gains Sobolev–Morrey regularity from the data of the problem.