Multi-peak solutions for super-critical elliptic problems in domains with small holes (Q1614755)

This paper is devoted to the construction of solutions of the problem \[ \begin{cases} -\Delta u=u^{{N+2 \over N-2}+\varepsilon} \quad &\text{in } \Omega,\\ u>0\quad &\text{in }\Omega,\\ u=0 \quad &\text{on }\partial\Omega, \end{cases}\tag{1} \] where \(\Omega\) is a bounded domain with smooth boundary in \(\mathbb{R}^N\), \(N\geq 3\), and \(\varepsilon >0\) is a small parameter. The goal of this paper is to raise the issue of solvability (1) in a domain exhibiting multiple holes. The main result of the authors states that in such a situation, multi-peak solutions exist, consisting of the glueing of double-spikes associated with each of the holes.