Let Ω ⊂ R n be a bounded smooth domain. For k > 1, consider the following non-linear elliptic boundary value problem (I)(÷ A l (x, u, D u) + f l (x, u, D u) = 0 i n Ω, l = 1, ..., k,; u (x) = u 0 (x), o n ∂ Ω,) where (u : Ω → R k , f l : Ω × R k × R n k → R; A l : Ω × R k × R n k → R n , l = 1, ..., k,) are vector-valued functions. Under appropriate conditions on the functions f l and A l , we investigate the question of existence of positive solutions to the boundary value problem (I) via the fixed point theory. The a priori estimates play a key role. © 2008 Elsevier B.V. All rights reserved.