On the quasilinear elliptic problems with critical Sobolev–Hardy exponents and Hardy terms

Abstract

Let Ω ⊂ R N be a smooth bounded domain such that 0 ∈ Ω , N ≥ 3 . In this paper, we study the critical quasilinear elliptic problems − 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 , p ≤ q < p ∗ ( s ) ≔ p ( N − s ) N − p , p ∗ ( t ) ≔ p ( N − t ) N − p , p ∗ ( s ) and p ∗ ( t ) are the critical Sobolev–Hardy exponents. Via variational methods, we deal with the conditions that ensure the existence of positive solutions for the equation. The results depend crucially on the parameters p , q , s , λ and μ .