On the elliptic problems with critical weighted Sobolev–Hardy exponents

Let Ω ⊂ R N be a smooth bounded domain such that 0 ∈ Ω , N ≥ 3 . In this paper, we deal with the conditions that ensure the existence of nontrivial solutions for the elliptic equation − div ( | x | − 2 a ∇ u ) − μ u | x | 2 ( 1 + a ) = | u | p − 2 | x | b p u + λ u with Dirichlet boundary condition, which involving the Caffarelli–Kohn–Nirenberg inequalities. The results depend crucially on the parameters a , b , λ and μ .