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 the 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,\ u_i = 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, $ $ \lambda_i $ are positive small numbers, $ \beta_{ij} $ are real numbers so that $ \beta_{ii} &gt; 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} &gt; 0 $, while forces among components of different groups are repulsive or weakly attractive, i.e., $ \beta_{ij} &lt; \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.</p></abstract>

The compactness of minimizing sequences for a nonlinear Schrödinger system with potentials

In this paper, we consider the following minimizing problem with two constraints: [Formula: see text] where [Formula: see text] and [Formula: see text] is defined by [Formula: see text] [Formula: see text] Here [Formula: see text], [Formula: see text] and [Formula: see text] [Formula: see text] are given functions. For [Formula: see text], we consider two cases: (i) both of [Formula: see text] and [Formula: see text] are bounded, (ii) one of [Formula: see text] and [Formula: see text] is bounded. Under some assumptions on [Formula: see text] and [Formula: see text], we discuss the compactness of any minimizing sequence.