Multiple solutions to a weakly coupled purely critical elliptic system in bounded domains

TL;DR

This paper investigates multiple solutions to a critical elliptic system with weak coupling in bounded domains, establishing existence results under symmetry conditions and analyzing phase separation as the coupling parameter varies.

Contribution

It introduces new existence results for multiple solutions to a critical elliptic system with symmetry assumptions and studies phase separation behavior for large negative coupling.

Findings

Existence of multiple fully nontrivial solutions under symmetry conditions.

Positive least energy solutions exhibit phase separation as   -.

Infinitely many solutions exist in ^N.

Abstract

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 $\Omega=\mathbb{R}^N$.