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

<abstract><p>We study the existence of solutions to cubic Schrödinger system</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -\Delta u_i = \sum\limits_{j 1}^m \beta_{ij} u_j^2u_i + \lambda_i u_i\ \hbox{in}\ \Omega,\ 0\ \hbox{on}\ \partial\Omega,\ i 1,\dots,m, $\end{document} </tex-math></disp-formula></p> <p>when $ \Omega is a bounded domain in \mathbb R^4, are positive small numbers, real numbers so that \beta_{ii} > 0 and \beta_{ji} $, i\neq j $. We assemble components groups all interaction forces among same group attractive, i.e., while different repulsive or weakly < \overline\beta for some small. find such each component within given blows-up around point blow-up points, as parameters $'s approach zero.</p></abstract>