A fountain of positive Bubbles on a Coron's Problem for a Competitive Weakly Coupled Gradient System

Abstract

We consider the following critical elliptic system: \begin{equation*} \begin{cases} -Δu_i=μ_i u_i^{3}+βu_i^{ } \sum\limits_{j\neq i} u_j^{2} \quad \hbox{in}\ Ω_\varepsilon \\ u_i=0 \hbox{ on } \partialΩ_\varepsilon , \qquad u_i>0 \hbox{ in } Ω_\varepsilon \end{cases}\qquad i=1,\ldots, m, \end{equation*} in a domain $Ω_\varepsilon \subset \mathbb{R}^4$ with a small shrinking hole $B_\varepsilon(ξ_0)$. For $μ_i>0$, $β<0$, and $\varepsilon>0$ small, we prove the existence of a non-synchronized solution which looks like a fountain of positive bubbles, i.e. each component $u_i$ exhibits a towering blow-up around $ξ_0$ as $\varepsilon \to 0$. 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 balance 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.