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We focus on the study of ground-states for the system of $M$ coupledsemilinear Schr\"odinger equations with power-type nonlinearities andcouplings. We extend the characterization result from a previous work(arXiv:1410.7993) to the case where both attraction and repulsion are presentand cannot be studied separately. Furthermore, we derive some perturbation andclassification results to study the general system where components may be outof phase. In particular, we present several conditions to the existence ofnontrivial ground-states.

Ground-states for systems of $M$ coupled semilinear Schr\"odinger  equations with attraction-repulsion effects: characterization and  perturbation results