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

Let Ω ⊂ R be a smooth bounded domain. In this paper we investigate the existence of least energy positive solutions to the following Schrödinger system with d ≥ 2 equations −∆ui + λiui = |ui| ui d ∑ j=1 βij |uj | p in Ω, ui = 0 on ∂Ω, i = 1, ..., d, in the case of a critical exponent 2p = 2 = 2N N−2 in high dimensions N ≥ 5. We treat the focusing case (βii > 0 for every i) in the variational setting βij = βji for every i 6= j, dealing with a Brézis-Nirenberg type problem: −λ1(Ω) < λi < 0, where λ1(Ω) is the first eigenvalue of (−∆, H 0 (Ω)). We provide several sufficient conditions on the coefficients βij that ensure the existence of least energy positive solutions; these include the situations of pure cooperation (βij > 0 for every i 6= j), pure competition (βij ≤ 0 for every i 6= j) and coexistence of both cooperation and competition coefficients. In order to provide these results, we firstly establish the existence of nonnegative solutions with 1 ≤ m ≤ d nontrivial components by comparing energy levels of the system with those of appropriate sub-systems. Moreover, based on new energy estimates, we obtain the precise asymptotic behaviours of the nonnegative solutions in some situations which include an analysis of a phase separation phenomena. Some proofs depend heavily on the fact that 1 < p < 2, revealing some different phenomena comparing to the special case N = 4. Our results provide a rather complete picture in the particular situation where the components are divided in two groups. Besides, based on the results about the phase separation, we prove the existence of least energy sign-changing solution to the Brézis-Nirenberg problem −∆u+ λu = μ|u| ∗ u, u ∈ H 0 (Ω), for μ > 0, −λ1(Ω) < λ < 0 for all N ≥ 4, a result which is new in dimensions N = 4, 5.