Infinitely many sign-changing solutions for the brézis-nirenberg problem involving hardy potential

Abstract In this article, we give a new proof on the existence of infinitely many sign-changing solutions for the following Brezis-Nirenberg problem with critical exponent and a Hardy potential − Δ u − μ u | x | 2 = λ u + | u | 2 * − 2 u in Ω , u = 0 on ∂ Ω , where Ω is a smooth open bounded domain of which contains the origin, 2 * = 2 N N − 2 is the critical Sobolev exponent. More precisely, under the assumptions that N ≥ 7 , μ ∈ [ 0 , μ ¯ − 4 ) , and μ ¯ = ( N − 2 ) 2 4 , we show that the problem admits infinitely many sign-changing solutions for each fixed λ > 0. Our proof is based on a combination of invariant sets method and Ljusternik-Schnirelman theory.