r

α + β |u| α−2u|v|β, x ∈ Ω −div ((γ + |∇v|r−2)∇v) = µ|v|r−2v + 2β α + β |u| α|v|β−2v, x ∈ Ω u = v = 0, x ∈ ∂Ω Variational setting Let Ω ⊂ RN be a smooth bounded domain, 2 ≤ p, r, γ ≥ 0, 1 ≤ α, β and N ≥ max{p2, r2}. Solutions of (P) correspond to critical points of the functional J: X ≡ W 1,p0 (Ω)×W 1,r0 (Ω) → R given by J(z) = J(u, v) =