Abstract:In this paper we prove a regularity result for very weak solutions of equations of the type $- div A(x,u,Du)=B(x, u,Du)$, where $A$, $B$ grow in the gradient like $t^{p-1}$ and $B(x, u, Du)$ is not in divergence form. Namely we prove that a very weak solution $u\in W^{1,r}$ of our equation belongs to $W^{1,p}$. We also prove global higher integrability for a very weak solution for the Dirichlet problem $$ \cases -div A(x,u,Du) =B(x, u,Du) \hskip 1em\relax & \text {in } \Omega , u-u_o\in W^{1,r}(\Omega ,\Bbb R^m). \endcases $$
Keywords: nonlinear elliptic systems, maximal operator theory
AMS Subject Classification: Primary 35J50, 35J55, 35J99; Secondary 46E30