In this article, we are concerned with the existence of positive radial solutions of the problem (S⁺): -Δ_pu= f(x,u,v) in Ω,
-Δ_qv= g(x,u,v) in Ω,
u = v = 0 on \partialΩ, where Ω is a ball in R^N and f, g are positive functions satisfying f(x,0,0)=g(x,0,0)=0. Under some growth conditions, we show the existence of a positive radial solution of the problem S⁺. We use traditional techniques of the topological degree theory. When Ω=R^N, we give some sufficient conditions of nonexistence.