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

## TL;DR

This paper constructs solutions with multiple positive bubbles concentrating at a point in a critical elliptic system with a small hole, revealing complex bubble interactions and blow-up behavior.

## Contribution

It introduces a new type of non-synchronized bubble solutions in a coupled elliptic system with a shrinking domain hole, using Lyapunov-Schmidt reduction.

## Key findings

- Existence of fountain-like positive bubble solutions
- Detailed analysis of bubble interaction balancing boundary effects
- Demonstration of complex blow-up phenomena in coupled systems

## Abstract

We consider the following critical elliptic system: \begin{equation*} \begin{cases} -\Delta u_i=\mu_i u_i^{3}+\beta u_i^{ } \sum\limits_{j\neq i} u_j^{2} \quad \hbox{in}\ \Omega_\varepsilon \\ u_i=0 \hbox{ on } \partial\Omega_\varepsilon , \qquad u_i>0 \hbox{ in } \Omega_\varepsilon \end{cases}\qquad i=1,\ldots, m, \end{equation*} in a domain $\Omega_\varepsilon \subset \mathbb{R}^4$ with a small shrinking hole $B_\varepsilon(\xi_0)$. For $\mu_i>0$, $\beta<0$, 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 $\xi_0$ 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.