Solutions to a cubic Schrödinger system with mixed attractive and repulsive forces in a critical regime

We study the existence of solutions to the cubic Schrödinger system

$$-\mathrm{\Delta }{u}_i=\sum _{j=1}^m{\beta }_{ij}{u}_j^2{u}_i+{\lambda }_i{u}_i\text{in}\mathrm{\Omega },u_i=0\text{on}\mathrm{\partial }\mathrm{\Omega },i=1,\dots ,m,$$

when $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^4,$ ${\lambda }_i$ are positive small numbers, ${\beta }_{ij}$ are real numbers so that ${\beta }_{ii}>0$ and ${\beta }_{ij}={\beta }_{ji}$, $i\ne j$. We assemble the components $u_i$ in groups so that all the interaction forces ${\beta }_{ij}$ among components of the same group are attractive, i.e., ${\beta }_{ij}>0$, while forces among components of different groups are repulsive or weakly attractive, i.e., ${\beta }_{ij}<\overline{\beta }$ for some $\overline{\beta }$ small. We find solutions such that each component within a given group blows-up around the same point and the different groups blow-up around different points, as all the parameters ${\lambda }_i$'s approach zero.