Higher differentiability results for solutions to a class of non-homogeneouns elliptic problems under sub-quadratic growth conditions

We prove a sharp higher differentiability result for local minimizers of functionals with non-autonomous integrand which is convex with respect to the gradientvariable, under p-growth conditions, with 1<2. The main novelty here is that the results are obtained assuming that the coefficients have weak derivatives in some Lebesgue space Lq and the datumfis assumed to belong to a suitable Lebesgue space Lr. We also prove that it is possible to weaken the assumption on the datum f and on the coefficients if the minimizers are assumed to be a priori bounded.