Existence of multiple solutions for an elliptic system with sign-changing weight functions

In this paper, we will consider a class of quasilinear elliptic problem of the form {−div(|x|−ap|∇u|p−2∇u)+g1(x)|u|p−2u=αα+βh(x)|u|α−2u|v|β+λH1(x)|u|n−2u,−div(|x|−ap|∇v|p−2∇v)+g2(x)|v|p−2v=βα+βh(x)|v|β−2v|u|α+μH2(x)|v|n−2v,u(x)>0,v(x)>0,x∈RN, where λλ, μ>0μ>0, 10d=a+1−b>0, the weight g1(x)g1(x), g2(x)g2(x) are bounded and nonnegative functions and h(x)h(x), H1(x)H1(x), H2(x)H2(x) are continuous functions which change sign in RNRN. We will prove that the problem has at least two positive solutions by using the Nehari manifold and the fibering maps associated with the Euler function for this problem.