Yamabe systems and optimal partitions on manifolds with symmetries

We prove the existence of regular optimal \begin{document}$ G $\end{document} -invariant partitions, with an arbitrary number \begin{document}$ \ell\geq 2 $\end{document} of components, for the Yamabe equation on a closed Riemannian manifold \begin{document}$ (M,g) $\end{document} when \begin{document}$ G $\end{document} is a compact group of isometries of \begin{document}$ M $\end{document} with infinite orbits. To this aim, we study a weakly coupled competitive elliptic system of \begin{document}$ \ell $\end{document} equations, related to the Yamabe equation. We show that this system has a least energy \begin{document}$ G $\end{document} -invariant solution with nontrivial components and we show that the limit profiles of its components separate spatially as the competition parameter goes to \begin{document}$ -\infty $\end{document} , giving rise to an optimal partition. For \begin{document}$ \ell = 2 $\end{document} the optimal partition obtained yields a least energy sign-changing \begin{document}$ G $\end{document} -invariant solution to the Yamabe equation with precisely two nodal domains.