We prove the existence of regular optimal $ G $-invariant partitions, with an arbitrary number $ \ell\geq 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 $ \ell $ 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 $ -\infty $, giving rise to an optimal partition. For $ \ell = 2 $ the optimal partition obtained yields a least energy sign-changing $ G $-invariant solution to the Yamabe equation with precisely two nodal domains.
Citation: Mónica Clapp, Angela Pistoia. Yamabe systems and optimal partitions on manifolds with symmetries[J]. Electronic Research Archive, 2021, 29(6): 4327-4338. doi: 10.3934/era.2021088
We prove the existence of regular optimal $ G $-invariant partitions, with an arbitrary number $ \ell\geq 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 $ \ell $ 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 $ -\infty $, giving rise to an optimal partition. For $ \ell = 2 $ the optimal partition obtained yields a least energy sign-changing $ G $-invariant solution to the Yamabe equation with precisely two nodal domains.
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