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 $N=4,5$.
Categories: math.AP