Existence of multiple solutions for a singular quasilinear elliptic equation in [formula omitted

Abstract

In this paper, we are concerned with the existence of multiple positive solutions for the singular quasilinear elliptic problem in R N { − div ( | x | − a p | ∇ u | p − 2 ∇ u ) + g ( x ) | u | p − 2 u = h ( x ) | u | m − 2 u + λ H ( x ) | u | n − 2 u , x ∈ R N u ( x ) > 0 , x ∈ R N . where λ > 0 is a real parameter and 1 < p < N ( N ≥ 3 ) , 1 < n < p < m < p ∗ , 0 ≤ a < ( N − p ) / p , p ∗ = N p / ( N − p d ) , a ≤ b < a + 1 , d = a + 1 − b > 0 . The weight function g ( x ) is a bounded nonnegative function with ‖ g ‖ ∞ > 0 and h ( x ) , H ( x ) are continuous functions which change sign in R N . We prove that there admits at least two positive solutions by using the Nehari manifold and the fibrering maps associated with the Euler functional for this problem.