A note on the existence of positive solution for a class of Laplacian nonlinear system with sign-changing weight

This study concerns the existence of positive solution for the system  −∆u = λ a(x) f(v), x ∈ Ω, −∆v = λ b(x) g(u), x ∈ Ω, u = v = 0, x ∈ ∂Ω, where λ > 0 is a parameter, Ω is a bounded domain in R(N > 1) with smooth boundary ∂Ω and ∆ is the Laplacian operator. Here a(x) and b(x) are C sign-changing functions that maybe negative near the boundary and f , g are C nondecresing functions such that f, g : [0,∞) → [0,∞) ; f(s), g(s) > 0 ; s > 0 and lim x→∞ f(Mg(x)) x = 0, 339 S.H. Rasouli, Z. Halimi and Z. Mashhadban/ TJMCS Vol .3 No.3 (2011) 339 345