Least energy positive solutions of critical Schrödinger systems with mixed competition and cooperation terms: The higher dimensional case

Let ??RN be a smooth bounded domain. In this paper we investigate the existence of least energy positive solutions to following Schrödinger system with d?2 equations??ui+?iui=|ui|p?2ui?j=1d?ij|uj|p in ?,ui=0 on ??,i=1,...,d, case critical exponent 2p=2?=2NN?2 high dimensions N?5. We treat focusing (?ii>0 for every i) variational setting ?ij=?ji i?j, dealing Brézis-Nirenberg type problem: ??1(?)0 i?j), competition (?ij?0 i?j) and coexistence both coefficients. order results, firstly establish nonnegative 1?m?d nontrivial components by comparing levels those appropriate sub-systems. Moreover, based new estimates, obtain precise asymptotic behaviors some which an analysis phase separation phenomena. Some proofs depend heavily fact 10, ??1(?)<?<0 all N?4, result N=4,5.