Elliptic systems with nonlinear diffusion and a convection term

In this paper we prove existence (and summability properties) of solutions for the following elliptic system\begin{document}$ \left\{ \begin{array}{cl} -{\rm{div}}(A(x)\,{\nabla} u) + u^{{\lambda}} = -{\rm{div}}( u^{{\lambda}} \, M(x)\,{\nabla}\psi) + f(x)\,, & {\rm{in }}\; \Omega , \\ -{\rm{div}}(M(x)\,{\nabla}\psi) = u^{\rho}\,, & {\rm{in }}\; \Omega , \\ u = 0 = \psi & \;{\rm{on}}\; \partial\Omega , \end{array} \right. $\end{document}under some assumptions on \begin{document}$ {\lambda} > 0 $\end{document} , \begin{document}$ \rho > 0 $\end{document} and \begin{document}$ f(x) $\end{document} in \begin{document}$ L^{{m}}(\Omega) $\end{document} , \begin{document}$ m \geq 1 $\end{document} . "Return of the Patriarca" (see [ 3 ] )