Higher differentiability and integrability for some nonlinear elliptic systems with growth coefficients in BMO

We consider local solutions u of nonlinear elliptic systems of the type aligned div \,A(x, Du) = div \, F in Ω ⊂ Rⁿ, aligned div A ( x , D u ) = div F in Ω ⊂ R n , where u: Ω → RN u : Ω → R N is in a weighted W1, ploc W loc 1 , p space, with p ≥ 2 p ≥ 2 , F is in a weighted W1, 2loc W loc 1 , 2 space and x → → A(x, ξ ) A ( x , ξ ) has growth coefficients in the space of functions with bounded mean oscillation. We prove higher differentiability of u in the sense that the nonlinear expression of its gradient V_μ (Du):=(μ ² + |Du|²)p - 2/4Du V μ ( D u ) : = ( μ 2 + | D u | 2 ) p - 2 4 D u , with 0 < μ ≤ 1 0 < μ ≤ 1 , is weakly differentiable with D(V_μ (Du)) D ( V μ ( D u ) ) in a weighted L²loc L loc 2 space. Moreover we derive some local Calderón–Zygmund estimates when the source term is not necessarily differentiable. Global estimates for a suitable Dirichlet problem are also available.