On Coron's problem for weakly coupled elliptic systems

Abstract

We consider the following critical weakly coupled elliptic system \[ \begin{cases} -Δu_i = μ_i |u_i|^{2^*-2}u_i + \sum_{j \neq i} β_{ij} |u_j|^{\frac{2^*}{2}} |u_i|^{\frac{2^*-4}{2}} u_i & \text{in ${\Omega }_{\epsilon }$} u_i >0 & \text{in ${\Omega }_{\epsilon }$} u_i = 0 & \text{on $\partial {\Omega }_{\epsilon }$},\end{cases} \qquad i =1,\dots,m, \] in a domain ${\Omega }_{\epsilon }\subset {\mathbb{R}}^N$, $N=3,4$, with small shrinking holes as the parameter $\epsilon \to 0$. 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.