On Coron's problem for weakly coupled elliptic systems

TL;DR

This paper investigates the existence and concentration behavior of positive solutions to a critical weakly coupled elliptic system in domains with shrinking holes, revealing solutions that concentrate around different points or groups as the holes diminish.

Contribution

It establishes the existence of positive solutions exhibiting specific concentration patterns in domains with small holes, extending understanding of elliptic systems with critical exponents.

Findings

Solutions concentrate around different holes or groups as the domain shrinks.

Existence of solutions with prescribed concentration behaviors.

Analysis applicable to domains with small geometric perturbations.

Abstract

We consider the following critical weakly coupled elliptic system \[ \begin{cases} -\Delta u_i = \mu_i |u_i|^{2^*-2}u_i + \sum_{j \neq i} \beta_{ij} |u_j|^{\frac{2^*}{2}} |u_i|^{\frac{2^*-4}{2}} u_i & \text{in $\Omega_\varepsilon$} u_i >0 & \text{in $\Omega_\varepsilon$} u_i = 0 & \text{on $\partial \Omega_\varepsilon$},\end{cases} \qquad i =1,\dots,m, \] in a domain $\Omega_\varepsilon \subset \mathbb{R}^N$, $N=3,4$, with small shrinking holes as the parameter $\varepsilon \to 0$. 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.