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

Abstract

We consider the elliptic system $$-\Delta u_i=u_i^3+\sum _{\genfrac{}{}{0ex}{}{j=1}{j\ne i}}^{q+1}{\beta }_{ij}{u}_i{u}_j^{2}in{\mathbb{R}}^4,i=1,\dots ,q+1.$$ 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.