We study the elliptic system

where Ω is a bounded domain in RN , N ≥ 3, κ1, κ2 ∈ R, µ1, µ2, λ > 0, α, β > 1, and α + β = p ≤ 2∗ := 2N /(N − 2). For p ∈ (2, 2∗) we establish the existence of a ground state and of a prescribed number of fully nontrivial solutions to this system for λ sufficiently large. If p = 2∗ and κ1, κ2 > 0 we establish the existence of a ground state for λ sufficiently large if, either N ≥ 5, or N = 4 and neither κ1 nor κ2 are Dirichlet eigenvalues of −∆ in Ω.