Critical polyharmonic systems and optimal partitions

TL;DR

This paper proves the existence of symmetric solutions to a coupled polyharmonic system, analyzes their limiting behavior as parameters tend to negative infinity, and characterizes the resulting optimal partition of the domain.

Contribution

It introduces a new approach to solving coupled polyharmonic equations with symmetry and describes the shape of the optimal partitions formed in the limit.

Findings

Existence of symmetric solutions to the system.

Limiting profiles form disjoint smooth domains.

These domains solve a nonlinear optimal partition problem.

Abstract

We establish the existence of solutions to a weakly-coupled competitive system of polyharmonic equations in 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 R^N. We give a detailed description of the shape of these domains.