On Coron's problem for weakly coupled elliptic systems

We consider the following critical weakly coupled elliptic system $\begin{cases} -Δu_i = μ_i |u_i|^{2^*-2}u_i + \sum_{j \neq i} β_{ij} |u_j|^{\frac{2^*}{2}} |u_i|^{\frac{2^*-4}{2}} u_i & \text{in $Ω_\varepsilon$} u_i >0 & \text{in $Ω_\varepsilon$} u_i = 0 & \text{on $\partial Ω_\varepsilon$},\end{cases} \qquad i =1,\dots,m,$ in a domain $Ω_\varepsilon \subset \mathbb{R}^N$, $N=3,4$, with small shrinking holes as the parameter $\varepsilon \to 0$. We prove the existence of positive solutions of two different types: either each density concentrates around a different hole, or we have groups of components such that all the components within a single group concentrate around the same point, and different groups concentrate around different points.

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