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