Existence of sign-changing solutions for semilinear singular elliptic equations with critical Sobolev exponents

We consider the semilinear elliptic problem − Δ u − μ u | x | 2 = f ( x , u ) + K ( x ) | u | 2 * − 2 u in Ω, u = 0 on ∂Ω, where 0 ∈ Ω is a smooth bounded domain in R N , N ⩾ 4 , 0 ⩽ μ < ( N − 2 ) 2 4 , 2 ∗ : = 2 N N − 2 is the critical Sobolev exponent, f ( x , ⋅ ) has subcritical growth at infinity, K ( x ) > 0 is continuous. We prove the existence of sign-changing solutions under different assumptions when Ω is a usual domain and a symmetric domain, respectively.