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} -Δu=μ_{1}|u|^{2^{*}-2}u+λα|u|^{α-2}|v|^βu & \text{in }Ω,\\ -Δv=μ_{2}|v|^{2^{*}-2}v+λβ|u|^α|v|^{β-2}v & \text{in }Ω,\\ u=v=0 & \text{on }\partialΩ, \end{cases} \end{equation*} where $\Omega$ is a bounded smooth domain in ${\mathbb{R}}^N$, $N\ge 3$, $2^{\ast }:=\frac{2N}{N-2}$ is the critical Sobolev exponent, ${\mu }_1,{\mu }_2>0$, $\alpha ,\beta >1$, $\alpha +\beta =2^{\ast }$ 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 $\Omega ={\mathbb{R}}^N$.