Difference methods for a nonlinear elliptic system of partial differential equations

This paper proves the existence and uniqueness of non-negative solutions to the Dirichlet problem associated with the nonlinear elliptic system \[ Δ u k = b k ∏ l = 1 m u l n ( l ) , k = 1 , . . . , m ( ∗ ) \Delta {u_k} = {b_k}\prod \limits _{l = 1}^m {u_l^{n\left ( l \right )}} ,k = 1,...,m\left ( * \right ) \] where the b k {b_k} and the Dirichlet data u k = φ k {u_k} = {\varphi _k} are non-negative.