Nodal cluster solutions for the Brezis–Nirenberg problem in dimensions \(N\ge 7\)

We show that the classical Brezis–Nirenberg problem \(\begin{aligned} \Delta u + |u|^{4 \over N-2} u + \varepsilon u = 0,\quad {\text{ in }} \quad \Omega , \quad u= 0, \quad {\text{ on }} \quad \partial \Omega \end{aligned}\) $$\begin{array}{c}\hfill \mathrm{\Delta }u+{|u|}^{\frac{4}{N-2}}u+\epsilon u=0,\text{in}\mathrm{\Omega },u=0,\text{on}\partial \mathrm{\Omega }\end{array}$$ admits nodal solutions clustering around a point on the boundary of \(\Omega \) $\mathrm{\Omega }$ as \(\varepsilon \rightarrow 0\) $\epsilon \to 0$ , for smooth bounded domains \(\Omega \subset {\mathbb {R}^N}\) $\mathrm{\Omega }\subset {\mathbb{R}}^N$ in dimensions \(N\ge 7\) $N\ge 7$ .