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.