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

  • Mónica Clapp

    Universidad Nacional Autónoma de México, Querétaro, Mexico
  • Angela Pistoia

    Università di Roma “La Sapienza”, Roma, Italy
  • Hugo Tavares

    Universidade de Lisboa, Lisboa, Portugal
Yamabe systems, optimal partitions and nodal solutions to the Yamabe equation cover

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Abstract

We give conditions for the existence of regular optimal partitions, with an arbitrary number of components, for the Yamabe equation on a closed Riemannian manifold . 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 , is not locally conformally flat, and satisfies an additional geometric assumption whenever . 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 the optimal partition obtained yields a least energy sign-changing solution to the Yamabe equation with precisely two nodal domains.

Cite this article

Mónica Clapp, Angela Pistoia, Hugo Tavares, Yamabe systems, optimal partitions and nodal solutions to the Yamabe equation. J. Eur. Math. Soc. (2024), published online first

DOI 10.4171/JEMS/1439