  Let $\Omega\subset \mathbb{R}^{N}$ 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 $d\geq 2$ 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 $2p=2^*=\frac{2N}{N-2}$ in high dimensions $N\geq 5$. We treat the
focusing case ($\beta_{ii}>0$ for every $i$) in the variational setting
$\beta_{ij}=\beta_{ji}$ for every $i\neq j$, dealing with a Br\'ezis-Nirenberg
type problem: $-\lambda_{1}(\Omega)<\lambda_{i}<0$, where $\lambda_{1}(\Omega)$
is the first eigenvalue of $(-\Delta,H^1_0(\Omega))$. 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 $i\neq j$), pure competition
($\beta_{ij}\leq 0$ for every $i\neq j$) 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 \[ -\Delta u+\lambda
u=\mu |u|^{2^*-2}u,\quad u\in H^1_0(\Omega), \] for $\mu>0$,
$-\lambda_1(\Omega)<\lambda<0$ for all $N\geq 4$, a result which is new in
dimensions $N=4,5$.
