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

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