Critical polyharmonic systems and optimal partitions

Mónica Clapp; Juan Carlos Fernández; Alberto Saldaña

doi:10.3934/cpaa.2021141

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2021, Volume 20, Issue 11: 4007-4023. Doi: 10.3934/cpaa.2021141 open access

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Critical polyharmonic systems and optimal partitions

Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, 04510 Coyoacán, Ciudad de México, Mexico

^* Corresponding author
^* Corresponding author

Received: April 30, 2021

Revised: June 30, 2021

Early access: August 2021

Published: November 2021

M. Clapp was supported by CONACYT grant A1-S-10457 (Mexico), J.C. Fernández was supported by a CONACYT postdoctoral fellowship (Mexico), and A. Saldaña was supported by UNAM-DGAPA-PAPIIT grant IA101721 (Mexico)

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Abstract

We establish the existence of solutions to a weakly-coupled competitive system of polyharmonic equations in $ \mathbb{R}^N $ which are invariant under a group of conformal diffeomorphisms, and study the behavior of least energy solutions as the coupling parameters tend to $ -\infty $. We show that the supports of the limiting profiles of their components are pairwise disjoint smooth domains and solve a nonlinear optimal partition problem of $ \mathbb R^N $. We give a detailed description of the shape of these domains.

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Mathematics Subject Classification: Primary: 35G50; Secondary: 35G30, 35B06, 58J70, 35B33.

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