A fountain of positive bubbles on a Coron's problem for a competitive weakly coupled gradient system

We consider the following critical elliptic system: {−Δui=μiui3+βui∑j≠iuj2inΩεui=0 on ∂Ωε,ui>0 in Ωεi=1,…,m, in a domain Ωε⊂R4 with a small shrinking hole Bε(ξ0). For μi>0, β<0, and ε>0 small, we prove the existence of a non-synchronized solution which looks like a fountain of positive bubbles, i.e. each component ui exhibits a towering blow-up around ξ0 as ε→0. The proof is based on the Ljapunov-Schmidt reduction method, and the velocity of concentration of each layer within a given tower is chosen in such a way that the interaction between bubbles of different components balances the interaction of the first bubble of each component with the boundary of the domain, and in addition is dominant when compared with the interaction of two consecutive bubbles of the same component.