The paper deals with the critical weakly coupled elliptic system \[ \begin{cases} -\Delta u_{i}= \mu_i|u_i|^{2^*-2}u_i+\sum_{j\neq i}\beta_{ij}|u_j|^{\frac{2}{2}}|u_i|^{\frac{2^*-4}{2}}u_i &\quad \text{ in }\;\Omega_\epsilon,\\ u_i=0 &\quad \text{ on }\;\partial\Omega_\epsilon,\\ i= 1,\dots,m, \end{cases} \] where \(\Omega_\epsilon\subset \mathbb{R}^N\), \(N=3,4\), is a domain with small shrinking holes as \(\epsilon \to 0\).
The authors derive existence of positive solutions of two different types: either each density concentrates around a different hole, or groups of components exist such that all the components within a single group concentrate around the same point, and different groups concentrate around different points. The technique of proofs rely on a perturbation approach based on the Lyapunov-Schmidt finite-dimensional reduction.