The Nehari manifold and the existence of multiple solutions for a singular quasilinear elliptic equation

In this paper, we are concerned with the existence of multiple positive solutions for the singular quasilinear elliptic problem { − div ( | x | − a p | ∇ u | p − 2 ∇ u ) = λ h ( x ) | u | m − 2 u + H ( x ) | u | n − 2 u , x ∈ Ω , u ( x ) = 0 , x ∈ ∂ Ω , where Ω ⊂ R N ( N ≥ 3 ) is a bounded domain with smooth boundary ∂ Ω , 0 ∈ Ω , 1 < p < N , 0 ≤ a < ( N − p ) / p , 1 < m < p < n < p N / ( N − ( 1 + a ) p ) , λ > 0 . h ( x ) , H ( x ) are Lebesgue measurable functions which may change sign on Ω . We prove that there exist at least two positive solutions by using the Nehari manifold and the fibrering maps associated with the energy functional for this problem.