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 ≥ 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 equations, related to the Yamabe equation. We show that this system has a least energy solution with nontrivial components if M≥ 10 , (M,g) is not locally conformally flat, and satisfies an additional geometric assumption whenever M=10 . Moreover, we show that the limit profiles of the components of the solution separate spatially as the competition parameter goes to -∞ , 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 =2 the optimal partition obtained yields a least energy sign-changing solution to the Yamabe equation with precisely two nodal domains.