Multiple positive solutions for a quasilinear elliptic problem involving critical Sobolev–Hardy exponents and concave–convex nonlinearities

Abstract

Let Ω ⊂ R N ( N ≥ 3 ) be a bounded smooth domain containing the origin. In this paper, by using variational methods, the multiplicity of positive solutions is obtained for a quasilinear elliptic problem − Δ p u − μ | u | p − 2 u | x | p = | u | p ∗ ( t ) − 2 | x | t u + λ | u | q − 2 | x | s u , u ∈ W 0 1 . p ( Ω ) , with Dirichlet boundary condition, where Δ p u = div ( | ∇ u | p − 2 ∇ u ) , 1 < p < N , 0 ≤ μ < μ ̄ = ( N − p p ) p , λ > 0 , 0 ≤ s , t < p , 1 ≤ q < p and p ∗ ( t ) = p ( N − t ) N − p is the critical Sobolev–Hardy exponent.