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

Abstract

AbstractWe show that the classical Brezis–Nirenberg problem $$\begin{aligned} \Delta u + |u|^{4 \over N-2} u + \varepsilon u = 0,\quad {\text{ in }} \quad \Omega , \quad u= 0, \quad {\text{ on }} \quad \partial \Omega \end{aligned}$$ Δ u + | u | 4 N - 2 u + ε u = 0 , in Ω , u = 0 , on ∂ Ω admits nodal solutions clustering around a point on the boundary of $$\Omega $$ Ω as $$\varepsilon \rightarrow 0$$ ε → 0 , for smooth bounded domains $$\Omega \subset {\mathbb {R}^N}$$ Ω ⊂ R N in dimensions $$N\ge 7$$ N ≥ 7 .