Segregated solutions for a critical elliptic system with a small interspecies repulsive force

TL;DR

This paper constructs solutions for a critical elliptic system in four dimensions with small interspecies repulsive forces, showing that multiple components can blow up at polygon vertices while a last component remains radial.

Contribution

It introduces a novel method to build segregated solutions with blow-up patterns in a coupled elliptic system under small negative interspecies interactions.

Findings

Solutions exhibit blow-up at polygon vertices on different great circles.

The last component remains similar to a radial positive solution.

The approach handles small negative coupling constants effectively.

Abstract

We consider the elliptic system $$-\Delta u_i = u_i^3+\sum\limits_{j=1\atop j\not=i}^{q+1}{ \beta_{ij}}u_i u_j^2\ \hbox{in}\ \mathbb R^4, \ i=1,\dots,q+1.$$ when $\alpha:=\beta_{ij}$ and $\beta:=\beta_{i(q+1)}=\beta_{(q+1)j}$ for any $i,j=1,\dots,q.$ If $\beta<0$ and $|\beta|$ is small enough we build solutions such that each component $u_{1},\dots,u_q$ blows-up at the vertices of $q$ polygons placed in different great circles which are linked to each other, and the last component $u_{q+1}$ looks like the radial positive solution of the single equation.