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

We consider the Brezis-Nirenberg problem: −∆u = λu + |u|^{2^*-2}u in Ω u = 0 on partial Ω where Ω is a smooth bounded domain in R^N , N ≥ 3, 2^∗ = 2N/(N-2) is the critical Sobolev exponent and λ > 0 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 form u_λ =PU_{δ_1,ξ} −PU_{δ_2,ξ} +w_λ, where PU_{δ_i} is the projection on H_0^1(Ω) of the regular positive solution of the critical problem in R^N , centered at a point ξ ∈ Ω and w_λ is a remainder term. Some additional results on norm estimates of w_λ and about the concentrations speeds of tower of bubbles in higher dimensions are also presented.