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

Abstract

We establish the existence of a positive fully nontrivial solution $(u,v)$ to the weakly coupled elliptic system% \[ \left\{ \begin{tabular} [c]{l}% $-\Delta u={\mu }_1|u{|}^{2^{\ast }-2}u+\lambda \alpha |u{|}^{\alpha -2}|v{|}^{\beta }u,$\\ $-Δv=μ_{2}|v|^{{2}^{\ast}-2}v+λβ|u|^α|v|^{β{-2}% }v,$\\ $u,v\in D^{1,2}({\mathbb{R}}^N),$% \end{tabular} \ \right. \] where $N\ge 4,$ $2^{\ast }:=\frac{2N}{N-2}$ 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 $\lambda \to -\infty ,$ 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.