Nodal solutions of the Brezis-Nirenberg problem in dimension 6

Abstract

We show that the classical Brezis-Nirenberg problem $$-\Delta u=u|u|+\lambda uin\Omega ,u=0on\partial \Omega ,$$ 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 ${\lambda }_0>0.$