Existence and non-existence of solutions for an elliptic system

We study the existence of positive solutions for a system of two elliptic equations of the form { − Δ u = a 1 ( x ) F 1 ( x , u , v ) in  Ω − Δ v = a 2 ( x ) F 2 ( x , u , v ) in  Ω u = v = 0 on  ∂ Ω where Ω ⊂ R N ( N ≥ 2 ) is a bounded domain in R N with a smooth boundary ∂ Ω or Ω = R N ( N ≥ 3 ) . A non-existence result is obtained for radially symmetric solutions. Our proofs are based primarily on the sub and super-solution method.