Let n ≥ 3 and Ω be either the entire space Rn or a Euclidean ball in Rn. Consider the following boundary value problem (Formula Presented) (I) with homogeneous Dirichlet boundary data (replaced by u, v → 0 as |x| → ∞ when Ω = Rn), where p > 1 and q > 1. In this paper, we investigate the question of existence and non-existence of solutions of (I) and prove that (I) admits a solution if and only if 1/p + 1 + 1/q + 1 > n - 2/n. The existence on a ball and on Rn are established by a variational approach and an approximation argument respectively. The Pohozaev identity is used to show non-existence on Rn.