Summary: We consider an equation of the form \[ \begin{cases} Lu+\beta(u)\ni f(x,u) & \text{ in } \Omega; \\ u=0 & \text{ on } \partial\Omega, \end{cases} \] where \(\Omega\subset\mathbb{R}^ n(n\geq 1)\) is an open set with smooth boundary \(\partial\Omega\), \(L=\text{diag}(L_ 1,\dots,L_ N)\) \((N\geq 1)\) is a diagonal matrix of second order elliptic operators, \(\beta\) is an \(m\)-accretive graph in \(\mathbb{R}^ N\) and \(f:\Omega\times\mathbb{R}^ N\to\mathbb{R}^ N\) is a given Caratheodory function satisfying some growth condition. We prove an existence result for this system.