We study the existence of solutions for a class of nonuniformly degenerate elliptic systems in N, N ≥ 3, of the form
where hi ∈ L1loc(N), hi(x) ≧ γ0|x|α with α ∈ (0, 2) and γ0 > 0, i = 1, 2. The proofs rely essentially on a variant of the Mountain pass theorem (D. M. Duc, Nonlinear singular elliptic equations, J. Lond. Math. Soc. 40(2) (1989), 420–440) combined with the Caffarelli–Kohn–Nirenberg inequality (First order interpolation inequalities with weights, Composito Math. 53 (1984), 259–275).