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 {equation presented} when Ω is a bounded domain in R4, λi are positive small numbers,βij are real numbers so thatβii > 0 andβij =βji, i ≠ j. We assemble the components ui in groups so that all the interaction forcesβij among components of the same group are attractive, i.e.,βij > 0, while forces among components of different groups are repulsive or weakly attractive, i.e.,βij