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

Abstract

We consider the Brezis–Nirenberg problem:{−Δu=λu+|u|2⁎−2uinΩ,u=0on∂Ω, where Ω is a smooth bounded domain in RN, N≥3, 2⁎=2NN−2 is the critical Sobolev exponent and λ>0 is a positive parameter.The main result of the paper shows that if N=4,5,6 and λ is close to zero, there are no sign-changing solutions of the formuλ=PUδ1,ξ−PUδ2,ξ+wλ, where PUδi is the projection on H01(Ω) of the regular positive solution of the critical problem in RN, centered at a point ξ∈Ω and wλ is a remainder term.Some additional results on norm estimates of wλ and about the concentrations speeds of tower of bubbles in higher dimensions are also presented.