Existence and phase separation of entire solutions to a pure critical competitive elliptic system

We establish the existence of a positive fully nontrivial solution $(u,v)$ to the weakly coupled elliptic system% \left\{ \begin{tabular} [c]{l}% $-\Delta u=\mu_{1}|u|^{{2}^{\ast}-2}u+\lambda\alpha|u|^{\alpha-2}|v|^{\beta }u,$\\ $-\Delta v=\mu_{2}|v|^{{2}^{\ast}-2}v+\lambda\beta|u|^{\alpha}|v|^{\beta{-2}% }v,$\\ $u,v\in D^{1,2}(\mathbb{R}^{N}),$% \end{tabular} \ \right. where $N\geq4,$ $2^{\ast}:=\frac{2N}{N-2}$ is the critical Sobolev exponent, $\alpha,\beta\in(1,2],$ $\alpha+\beta=2^{\ast},$ $\mu_{1},\mu_{2}>0,$ and $\lambda<0.$ We show that these solutions exhibit phase separation as $\lambda\rightarrow-\infty,$ and we give a precise description of their limit domains. If $\mu_{1}=\mu_{2}$ and $\alpha=\beta$, we prove that the system has infinitely many fully nontrivial solutions, which are not conformally equivalent.