Yamabe systems, optimal partitions, and nodal solutions to the Yamabe equation

We give conditions for the existence of regular optimal partitions, with an arbitrary number $\ell\geq 2$ of components, for the Yamabe equation on a closed Riemannian manifold $(M,g)$. To this aim, we study a weakly coupled competitive elliptic system of $\ell$ equations, related to the Yamabe equation. We show that this system has a least energy solution with nontrivial components if $\dim M\geq 10$, $(M,g)$ is not locally conformally flat and satisfies an additional geometric assumption whenever $\dim M=10$. Moreover, we show that the limit profiles of the components of the solution separate spatially as the competition parameter goes to $-\infty$, giving rise to an optimal partition. We show that this partition exhausts the whole manifold, and we prove the regularity of both the interfaces and the limit profiles, together with a free boundary condition. For $\ell=2$ the optimal partition obtained yields a least energy sign-changing solution to the Yamabe equation with precisely two nodal domains.