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

Let Ω⊂RN 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−Δui+λiui=|ui|p−2ui∑j=1dβij|uj|p in Ω,ui=0 on ∂Ω,i=1,...,d, in the case of a critical exponent 2p=2⁎=2NN−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≠j, dealing with a Brézis-Nirenberg type problem: −λ1(Ω)<λi<0, where λ1(Ω) is the first eigenvalue of (−Δ,H01(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≠j), pure competition (βij≤0 for every i≠j) and coexistence of both cooperation and competition coefficients.