Existence of solutions for elliptic equations with critical Sobolev–Hardy exponents

Abstract

Let Ω⊂ R N be a smooth bounded domain such that 0∈Ω, N⩾3,0⩽s<2, 2 ∗(s)≔2(N−s)/(N−2),2<r<2 ∗≔2N/(N−2) . Via the variational methods, We prove the existence of sign-changing solutions for the singular critical problem − Δu−μu/|x| 2=|u| 2 ∗(s)−2 /|x| su+λ|u| r−2u with Dirichlet boundary condition on Ω for suitable positive parameters λ and μ.