Positive Solutions for a Class of p-Laplacian Systems with Sign-Changing Weight

We consider the system ⎧ ⎨ ⎩ −Δ p u = λF (x, u, v), x ∈ Ω, −Δ q v = λH(x, u, v), x ∈ Ω, u = 0 = v, x ∈ ∂Ω, where F (x, u, v) = [g(x)a(u) + f (v)], H(x, u, v) = [g(x)b(v) + h(u)], Ω is a bounded domain in R N (N ≥ 1) with smooth boundary ∂Ω, λ is a real positive parameter and Δ s z = div (|∇z| s−2 ∇z), s > 1, (s = p, q) is a s-laplacian operator. Here g is a C 1 sign-changing function that may be negative near the boundary and f, h, a, b are C 1 nondecreasing functions satisfying a(0) ≥ 0, b(0) ≥ 0, lim x→∞ a(x) x p−1 = 0, lim x→∞ b(x) x q−1 = 0, lim x→∞ f (x) = ∞, lim x→∞ h(x) = ∞, and lim x→∞ f (M [h(x)] 1 q−1) x p−1 = 0, ∀M > 0, by applying the method of sub-super solutions the existence of a positive solution is established for the above system.