Nodal Solutions of the Brezis-Nirenberg Problem in Dimension 6

Abstract

We show that the classical Brezis-Nirenberg problem $$-\Delta u=u|u|+\lambda u \ \ \ \ \ \  \ in \  \ \  \Omega, \\ u=0   \  \  \   \  \  \   \  \ \ \ \  \ \ \ \  \  \ \  \  \ \    \  \ \ on \   \  \  \partial\Omega,$$ when $\Omega$ is a bounded domain in $\mathbb R^6$ has a sign-changing solution which blows-up at a point in $\Omega$ as $\lambda$ approaches a suitable value $\lambda_0>0.$