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% \[ \left\{ \begin{tabular} [c]{l}% $-Δu=μ_{1}|u|^{{2}^{\ast}-2}u+λα|u|^{α-2}|v|^{β}u,$\\ $-Δv=μ_{2}|v|^{{2}^{\ast}-2}v+λβ|u|^α|v|^{β{-2}% }v,$\\ $u,v\in D^{1,2}(\mathbb{R}^{N}),$% \end{tabular} \ \right. \] where $N\geq4,$ $2^{\ast}:=\frac{2N}{N-2}$ is the critical Sobolev exponent, $α,β\in(1,2],$ $α+β=2^{\ast},$ $μ_{1},μ_{2}>0,$ and $λ<0.$ We show that these solutions exhibit phase separation as $λ\rightarrow-\infty,$ and we give a precise description of their limit domains. If $μ_{1}=μ_{2}$ and $α=β$, we prove that the system has infinitely many fully nontrivial solutions, which are not conformally equivalent.