Nodal cluster solutions for the Brezis-Nirenberg problem in dimensions $N\ge 7$

Abstract

We show that the classical Brezis-Nirenberg problem $$Δu + |u|^{4 \over N-2} u + \varepsilon u = 0 ,\quad {\mbox {in}} \quad Ω, \quad u= 0 , \quad {\mbox {on}} \quad \partial Ω$$ admits nodal solutions clustering around a point on the boundary of $\Omega$ as $\epsilon \to 0$, for smooth bounded domains $\Omega \subset {\mathbb{R}}^N$ in dimensions $N\ge 7$.