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 $ \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 $.