Solutions to a cubic Schr\"odinger system with mixed attractive and repulsive forces in a critical regime

Abstract

We study the existence of solutions to the cubic Schr\"odinger system $$ -\Delta u_i = \sum_{j =1}^m \beta_{ij} u_j^2u_i + \lambda_i u_i\ \hbox{in}\ \Omega,\ u_i=0\ \hbox{on}\ \partial\Omega,\ i =1,\dots,m, $$ 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.