On the elliptic problems with critical weighted Sobolev–Hardy exponents

Let Ω⊂RNΩ⊂RN be a smooth bounded domain such that 0∈Ω0∈Ω, N≥3N≥3. In this paper, we deal with the conditions that ensure the existence of nontrivial solutions for the elliptic equation −div(|x|−2a∇u)−μu|x|2(1+a)=|u|p−2|x|bpu+λu with Dirichlet boundary condition, which involving the Caffarelli–Kohn–Nirenberg inequalities. The results depend crucially on the parameters a,b,λa,b,λ and μμ.