We study the elliptic system −Δu = δ(x)^−av^p in Ω, −Δv = δ(x)^−bu^q in Ω, subject to homogeneous Dirichlet boundary conditions. Here, Ω ⊂ ℝ^N, N ≥ 1, is a smooth and bounded domain, δ(x) = dist(x, ∂Ω), a, b ≥ 0 and p, q ∈ ℝ satisfy pq > −1. The existence, non-existence and uniqueness of solutions are investigated in terms of a, b, p and q.