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

Abstract

We consider the Brezis-Nirenberg problem: \begin{equation*} \begin{cases} -Δu = λu + |u|^{2^* -2}u & \hbox{in}\ Ω\\ u=0 & \hbox{on}\ \partial Ω, \end{cases} \end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^N$, $N\geq 3$, $2^{*}=\frac{2N}{N-2}$ 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 $\mathbb{R}^N$, centered at a point $ξ\in Ω$ 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.