A nonexistence result for sign-changing solutions of the Brezis-Nirenberg problem in low dimensions

We consider the Brezis–Nirenberg problem: −Delta u = λu + |u|2∗−2u in Ω, u =0 on ∂Ω, where Ωis a smooth bounded domain in RN, N≥3, 2∗=2N/N−2is the critical Sobolev exponent and λ >0is a positive parameter. The main result of the paper shows that if N=4, 5, 6and λ is close to zero,there are no sign-changing solutions of the form uλ = PUδ1,ξ −PUδ2,ξ +wλ, where PUδiis the projection on H10(Ω)of the regular positive solution of the critical problem in RN, centered at a point ξ∈Ωand wλis a remainder term. Some additional results on norm estimates are given.