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

Abstract

We consider the Brezis-Nirenberg problem: $$-Δu =λu + |u|^{p-1}u\qquad \mbox{in}\,\, Ω,\quad u=0\,\, \mbox{on}\,\,\ \partialΩ,$$ where $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^N$, $N\ge 3$, $p=\frac{N+2}{N-2}$ 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 $\lambda \to {\lambda }_1$, where ${\lambda }_1={\lambda }_1(\Omega )$ denotes the first eigenvalue of $-\Delta$ on $\Omega$, with zero Dirichlet boundary condition.

arXiv admin note: substantial text overlap with arXiv:1402.1451