Yamabe systems and optimal partitions on manifolds with symmetries

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