Using variational methods, we study the existence and nonexistence of nontrivial weak solutions for the quasilinear elliptic system ⎧⎪⎨ ⎪⎩ −div(h1(|∇u|2)∇u) = μ |x|2 u + λFu(x, u, v) inΩ, −div(h2(|∇v|2)∇v) = μ |x|2 v + λFv(x, u, v) inΩ, u = v = 0 in ∂Ω, where Ω ⊂ RN,N ≥ 3, is a bounded domain containing the origin with smooth boundary ∂Ω; hi, i = 1, 2, are nonhomogeneous potentials; (Fu, Fv) = ∇F stands for the gradient of a sign-changing C1-function F : Ω × R2 → R in the variable w = (u, v) ∈ R2; and λ and μ are parameters.
