Sign-changing blowing-up solutions for the Brezis--Nirenberg problem in dimensions four and five

Abstract

We consider the Brezis-Nirenberg problem: $$-\Delta u =\lambda u + |u|^{p-1}u\qquad \mbox{in}\,\, \Omega,\quad u=0\,\, \mbox{on}\,\,\ \partial\Omega,$$ where $\Omega$ is a smooth bounded domain in $\mathbb R^N$, $N\geq 3$, $p=\frac{N+2}{N-2}$ and $\lambda>0$. In this paper we prove that, if $\Omega$ is symmetric and $N=4,5$, there exists a sign-changing solution whose positive part concentrates and blows-up at the center of symmetry of the domain, while the negative part vanishes, as $\lambda\rightarrow \lambda_1$, where $\lambda_1=\lambda_1(\Omega)$ denotes the first eigenvalue of $-\Delta$ on $\Omega$, with zero Dirichlet boundary condition.