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

$-\Delta u_i = \sum\limits_{j = 1}^m \beta_{ij} u_j^2u_i + \lambda_i u_i\ \hbox{in}\ \Omega,\ u_i = 0\ \hbox{on}\ \partial\Omega,\ i = 1,\dots,m,$

when $\Omega$ 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 $\beta_{ij} = \beta_{ji}$, $i\neq j$. 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.