Double-spike solutions for a critical inhomogeneous elliptic problem in domains with small holes

Abstract

In this paper we construct solutions which develop two negative spikes as $\varepsilon\to0^+$ for the problem $-\Delta u=|u|^{4/(N-2)}u+\varepsilon f(x)$ in $\varOmega$, $u=0$ on $\partial\varOmega$, where $\varOmega\subset\mathbb{R}^N$ is a bounded smooth domain exhibiting a small hole, with $f\geq0$, $f\not\equiv0$. This result extends a recent work of Clapp et al. in the sense that no symmetry assumptions on the domain are required.