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 $Ω$ is a bounded smooth domain in $\mathbb{R}^{N}$, $N\geq 3$, $2^{*}:=\frac{2N}{N-2}$ is the critical Sobolev exponent, $μ_{1},μ_{2}>0$, $α, β>1$, $α+β=2^{*}$ and $λ\in\mathbb{R}$. We establish the existence of a prescribed number of fully nontrivial solutions to this system under suitable symmetry assumptions on $Ω$, which allow domains with finite symmetries, and we show that the positive least energy symmetric solution exhibits phase separation as $λ\to -\infty$. We also obtain existence of infinitely many solutions to this system in $Ω=\mathbb{R}^N$.