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 $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^N$, $N\ge 3$, $2^{\ast }=\frac{2N}{N-2}$ is the critical Sobolev exponent and $\lambda >0$ a positive parameter. The main result of the paper shows that if $N=4,5,6$ and $\lambda$ is close to zero there are no sign-changing solutions of the form $$u_{\lambda }=P{U}_{{\delta }_1,\xi }-P{U}_{{\delta }_2,\xi }+w_{\lambda },$$ where $P{U}_{{\delta }_i}$ is the projection on $H_0^1(\Omega )$ of the regular positive solution of the critical problem in ${\mathbb{R}}^N$, centered at a point $\xi \in \Omega$ and $w_{\lambda }$ is a remainder term. Some additional results on norm estimates of $w_{\lambda }$ and about the concentrations speeds of tower of bubbles in higher dimensions are also presented.

21 pages