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 \(\begin{aligned} \text {div} \,A(x, Du) = \text {div} \, F \qquad \text {in} \quad \Omega \subset \mathbb {R}^n, \end{aligned}\) $$\begin{array}{c}\hfill \text{div}A(x,Du)=\text{div}F\text{in}\mathrm{\Omega }\subset {\mathbb{R}}^n,\end{array}$$ where \(u: \Omega \rightarrow \mathbb {R}^N\) $u:\mathrm{\Omega }\to {\mathbb{R}}^N$ is in a weighted \(W^{1, p}_{loc}\) $W_{\mathit{loc}}^{1,p}$ space, with \(p \ge 2\) $p\ge 2$ , F is in a weighted \(W^{1, 2}_{loc}\) $W_{\mathit{loc}}^{1,2}$ space and x \(\rightarrow \) $\to$ \(A(x, \xi )\) $A(x,\xi )$ 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 \(V_\mu (Du):=(\mu ^2 + |Du|^2)^{\frac{p - 2}{4}}Du\) $V_{\mu }(Du):=({\mu }^2+{|Du|}^2{)}^{\frac{p-2}{4}}Du$ , with \(0 < \mu \le 1\) $0<\mu \le 1$ , is weakly differentiable with \(D(V_\mu (Du))\) $D(V_{\mu }(Du))$ in a weighted \(L^2_{loc}\) $L_{\mathit{loc}}^2$ 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.