In this paper we investigate the existence of solutions to the following Schrödinger system in the critical caseHere, Ω ⊂ R 4 is a smooth bounded domain, d ≥ 2, −λ1(Ω) < λi < 0 and βii > 0 for every i, βij = βji for i = j, where λ1(Ω) is the first eigenvalue of −∆ with Dirichlet boundary conditions. Under the assumption that the components are divided into m groups, and that βij ≥ 0 (cooperation) whenever components i and j belong to the same group, while βij < 0 or βij is positive and small (competition or weak cooperation) for components i and j belonging to different groups, we establish the existence of nonnegative solutions with m nontrivial components, as well as classification results. Moreover, under additional assumptions on βij , we establish existence of least energy positive solutions in the case of mixed cooperation and competition. The proof is done by induction on the number of groups, and requires new estimates comparing energy levels of the system with those of appropriate sub-systems. In the case Ω = R 4 and λ1 = . . . = λm = 0, we present new nonexistence results. This paper can be seen as the counterpart of [Soave-Tavares, J. Differential Equations 261 (2016), 505-537] in the critical case, while extending and improving some results from [Chen-Zou, Arch.

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