Higher differentiability for a class of problems under p,q subquadratic growth

We study the higher differentiability for nonlinear elliptic equation in divergence form $\mathcal{A}(x,Du)=b(x)$. The result covers the cases in which $\mathcal{A}(x,\xi )$ satisfies $p,q$ growth, with $1<p<2$ in $\xi$ and a Sobolev dependence of with respect to $x$. By means of an a-priori estimate we ensure the $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2,p}(\mathrm{\Omega })$-property for the solution of the boundary value problem.