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

TL;DR

This paper proves that in low dimensions (4, 5, 6), there are no sign-changing solutions of a critical elliptic PDE near zero parameter, refining understanding of solution structures in these cases.

Contribution

It establishes a nonexistence result for sign-changing solutions of the Brezis-Nirenberg problem in low dimensions close to zero parameter, with detailed estimates and concentration analysis.

Findings

No sign-changing solutions for N=4,5,6 near zero lambda

Norm estimates for the remainder term w_lambda

Concentration speed analysis of bubble towers in higher dimensions

Abstract

We consider the Brezis-Nirenberg problem: \begin{equation*} \begin{cases} -\Delta u = \lambda u + |u|^{2^* -2}u & \hbox{in}\ \Omega\\ u=0 & \hbox{on}\ \partial \Omega, \end{cases} \end{equation*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $N\geq 3$, $2^{*}=\frac{2N}{N-2}$ 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 $$u_\lambda=PU_{\delta_1,\xi}-PU_{\delta_2,\xi}+w_\lambda, $$ where $PU_{\delta_i}$ 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.