Yamabe systems and optimal partitions on manifolds with symmetries

Abstract

<p style='text-indent:20px;'>We prove the existence of regular optimal <inline-formula><tex-math id="M1">\begin{document}$ G $\end{document}</tex-math></inline-formula>-invariant partitions, with an arbitrary number <inline-formula><tex-math id="M2">\begin{document}$ \ell\geq 2 $\end{document}</tex-math></inline-formula> of components, for the Yamabe equation on a closed Riemannian manifold <inline-formula><tex-math id="M3">\begin{document}$ (M,g) $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M4">\begin{document}$ G $\end{document}</tex-math></inline-formula> is a compact group of isometries of <inline-formula><tex-math id="M5">\begin{document}$ M $\end{document}</tex-math></inline-formula> with infinite orbits. To this aim, we study a weakly coupled competitive elliptic system of <inline-formula><tex-math id="M6">\begin{document}$ \ell $\end{document}</tex-math></inline-formula> equations, related to the Yamabe equation. We show that this system has a least energy <inline-formula><tex-math id="M7">\begin{document}$ G $\end{document}</tex-math></inline-formula>-invariant solution with nontrivial components and we show that the limit profiles of its components separate spatially as the competition parameter goes to <inline-formula><tex-math id="M8">\begin{document}$ -\infty $\end{document}</tex-math></inline-formula>, giving rise to an optimal partition. For <inline-formula><tex-math id="M9">\begin{document}$ \ell = 2 $\end{document}</tex-math></inline-formula> the optimal partition obtained yields a least energy sign-changing <inline-formula><tex-math id="M10">\begin{document}$ G $\end{document}</tex-math></inline-formula>-invariant solution to the Yamabe equation with precisely two nodal domains.</p>