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 {−Δu=a1(x)F1(x,u,v)in  Ω−Δv=a2(x)F2(x,u,v)in  Ωu=v=0on  ∂Ω where Ω⊂RN(N≥2)Ω⊂RN(N≥2) is a bounded domain in RNRN with a smooth boundary ∂Ω∂Ω or Ω=RN(N≥3)Ω=RN(N≥3). A non-existence result is obtained for radially symmetric solutions. Our proofs are based primarily on the sub and super-solution method.