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 { − d i v ( | x | − a p | ∇ u | p − 2 ∇ u ) + g 1 ( x ) | u | p − 2 u = α α + β h ( x ) | u | α − 2 u | v | β + λ H 1 ( x ) | u | n − 2 u , − d i v ( | x | − a p | ∇ v | p − 2 ∇ v ) + g 2 ( x ) | v | p − 2 v = β α + β h ( x ) | v | β − 2 v | u | α + μ H 2 ( x ) | v | n − 2 v , u ( x ) > 0 , v ( x ) > 0 , x ∈ R N , where λ , μ > 0 , 1 < p < N , 1 < n < p < α + β < p ∗ = N p N − p d , 0 ≤ a < N − p p , a ≤ b < a + 1 , d = a + 1 − b > 0 , the weight g 1 ( x ) , g 2 ( x ) are bounded and nonnegative functions and h ( x ) , H 1 ( x ) , H 2 ( x ) are continuous functions which change sign in R N . 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.