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

TL;DR

This paper proves the existence of positive solutions to a critical elliptic system with competitive interactions, describes their phase separation behavior as the interaction parameter becomes large negative, and finds infinitely many solutions under symmetric conditions.

Contribution

It establishes the existence of solutions to a critical elliptic system with competitive coupling and characterizes their phase separation and multiplicity properties.

Findings

Solutions exhibit phase separation as interaction becomes strongly negative.

Existence of infinitely many solutions when parameters are symmetric.

Precise description of solution limit domains during phase separation.

Abstract

We establish the existence of a positive fully nontrivial solution $(u,v)$ to the weakly coupled elliptic system% \[ \left\{ \begin{tabular} [c]{l}% $-\Delta u=\mu_{1}|u|^{{2}^{\ast}-2}u+\lambda\alpha|u|^{\alpha-2}|v|^{\beta }u,$\\ $-\Delta v=\mu_{2}|v|^{{2}^{\ast}-2}v+\lambda\beta|u|^{\alpha}|v|^{\beta{-2}% }v,$\\ $u,v\in D^{1,2}(\mathbb{R}^{N}),$% \end{tabular} \ \right. \] 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.