Existence and boundedness of weak solutions to some vectorial Dirichlet problems

We study the existence and boundedness of a weak solution $u$ of the following nonlinear vectorial Dirichlet problem \[ \left\{\begin{aligned} &u\in\W\\ & - \sum_{i=1}^n \, \Di A_i^\nu(x, \DD u)=- \sum_{i=1}^n \, \Di\Big(E_i^{\nu}(x)\,u^\nu\Big)+F^\nu (x)\quad \text{$x\in\Om$} \end{aligned}\right. \] where $\Om$ is a bounded subset of $\R^n$, $n\geq 3$, for any $\nu=1,2,\dots, N$ $u^\nu$ and $f^\nu$ are the $\nu-$th components of the vectors $u$ and $f$, respectively, and the tensor $A(x,\xi)$ satisfies suitable structural assumptions.