On Coron's problem for weakly coupled elliptic systems

We consider the following critical weakly coupled elliptic system -Δui=μi|ui|2-2ui+j≠iβij|uj|22|ui|2-42uiinΩϵui=0on∂Ωϵ,i=1,m,in a domain Ωϵ⊂ℝN, N=3,4, with small shrinking holes as the parameter ϵ→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.