Multiple solutions to a weakly coupled purely critical elliptic system in bounded domains

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 $\mathrm{\Omega }$ is a bounded smooth domain in ${\mathbb{R}}^N$, $N\ge 3$, $2^{\ast }:=\frac{2N}{N-2}$ is the critical Sobolev exponent, ${\mu }_1,{\mu }_2>0$, $\alpha ,\beta >1$, $\alpha +\beta =2^{\ast }$ 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 $\mathrm{\Omega }$, which allow domains with finite symmetries, and we show that the positive least energy symmetric solution exhibits phase separation as $\lambda \to -\mathrm{\infty }$. We also obtain existence of infinitely many solutions to this system in $\mathrm{\Omega }={\mathbb{R}}^N$.