On a class of quasilinear systems with sign-changing nonlinearities

We prove existence and nonexistence of positive solutions for the quasilinear system−Δpu=λa(x)f(u,v)inΩ,−Δqv=μb(x)g(u,v)inΩ,u=v=0on∂Ω, where Δpu=div(|∇u|p−2∇u), Δqv=div(|∇v|q−2∇v), p,q>1, Ω is a bounded domain in RN, and the coefficients a(x) and b(x) are allowed to change sign.