Least energy positive solutions of critical Schr\"{o}dinger systems with mixed competition and cooperation terms: the higher dimensional case

Let $\mathrm{\Omega }\subset {\mathbb{R}}^N$ be a smooth bounded domain. In this paper we investigate the existence of least energy positive solutions to the following Schr"{o}dinger system with $d\ge 2$ equations \begin{equation*} -\Delta u_{i}+\lambda_{i}u_{i}=|u_{i}|^{p-2}u_{i}\sum_{j = 1}^{d}\beta_{ij}|u_{j}|^{p} $\text{ in }$ \Omega, \quad u_i=0 $\text{ on }$ \partial \Omega, \qquad i=1,…,d, \end{equation*} in the case of a critical exponent $2p=2^{\ast }=\frac{2N}{N-2}$ in high dimensions $N\ge 5$. We treat the focusing case (${\beta }_{ii}>0$ for every $i$) in the variational setting ${\beta }_{ij}={\beta }_{ji}$ for every $i\mathrm{\ne }j$, dealing with a Br'ezis-Nirenberg type problem: $-{\lambda }_1(\mathrm{\Omega })<{\lambda }_i<0$, where ${\lambda }_1(\mathrm{\Omega })$ is the first eigenvalue of $(-\mathrm{\Delta },H_0^1(\mathrm{\Omega }))$. We provide several sufficient conditions on the coefficients ${\beta }_{ij}$ that ensure the existence of least energy positive solutions; these include the situations of pure cooperation (${\beta }_{ij}>0$ for every $i\mathrm{\ne }j$), pure competition (${\beta }_{ij}\le 0$ for every $i\mathrm{\ne }j$) and coexistence of both cooperation and competition coefficients. 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 a phase separation phenomena, we prove the existence of least energy sign-changing solution to the Br'ezis-Nirenberg problem $$-\mathrm{\Delta }u+\lambda u=\mu \mathrm{\mid }u{\mathrm{\mid }}^{2^{\ast }-2}u,u\in H_0^1(\mathrm{\Omega }),$$ for $\mu >0$, $-{\lambda }_1(\mathrm{\Omega })<\lambda <0$ for all $N\ge 4$, a result which is new in dimensions $N=4,5$.