We provide sharp estimates in Lorentz spaces for the solution of the Dirichlet problem associated to the system $$\left\{ \begin{array}{ll} A(u)\equiv-D_i (A_{ij}(x) D_j u)=f u \in W^{1,1}_{0}(\Omega, \mathbb {R}^N) \end{array} \right.$$ where Ω is an open bounded subset of \({\mathbb R^n}\) (n ≥ 3) with sufficiently regular boundary, A(u) is an elliptic operator with VMO-coefficients and f is not in the natural dual space. Moreover, when the coefficients belong to \({{{C}^{0,\alpha}}(\alpha\in ]0,1[)}\), we study the differentiability of the solution in Besov–Morrey spaces.