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 Brézis-Nirenberg problem with critical exponent and a Hardy potential −Δu−μu|x|2=λu+|u|2*−2u in Ω, u = 0 on ∂Ω,where Ω is a smooth open bounded domain of ▪ which contains the origin, 2*=2NN−2 is the critical Sobolev exponent. More precisely, under the assumptions that N≥7, μ∈[0,μ¯−4), and μ¯=(N−2)24, 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.