We study the weakly coupled critical elliptic system \begin{ equation*} \begin{ cases} -\Delta u=\mu_{ 1}|u|^{ 2^{ *}-2}u+\lambda\alpha |u|^{ \alpha-2}|v|^{ \beta}u & \text{ in }\Omega,\\ -\Delta v=\mu_{ 2}|v|^{ 2^{ *}-2}v+\lambda\beta |u|^{ \alpha}|v|^{ \beta-2}v & \text{ in }\Omega,\\ u=v=0 & \text{ on }\partial\Omega, \end{ cases} \end{ equation*} where $\Omega$ is a bounded smooth domain in $\mathbb{ R}^{ N}$, $N\geq 3$, $2^{ *}:=\frac{ 2N}{ N-2}$ is the critical Sobolev exponent, $\mu_{ 1},\mu_{ 2}>0$, $\alpha, \beta>1$, $\alpha+\beta =2^{ *}$ and $\lambda\in\mathbb{ R}$. We establish the existence of a prescribed number of fully nontrivial solutions to this system under suitable symmetry assumptions on $\Omega$, which allow domains with finite symmetries, and we show that the positive least energy symmetric solution exhibits phase separation as $\lambda\to -\infty$. We also obtain existence of infinitely many solutions to this system in $\Omega=\mathbb{ R}^N$.
Categories: math.AP