Entire solutions to equivariant elliptic systems with variational structure

TL;DR

This paper proves the existence of entire solutions to a class of equivariant elliptic systems with variational structure, connecting multiple global minima under symmetry constraints.

Contribution

It establishes the existence of nontrivial solutions for elliptic systems with symmetry and multiple minima, extending previous results to more general reflection group invariances.

Findings

Existence of solutions connecting global minima at infinity

Solutions respect symmetry under finite reflection groups

Applicable to systems with multiple potential minima

Abstract

In the present paper we consider the system {\Delta}u - W_u (u) = 0, where u: R^n to R^n, for a class of potentials W: R^n to R that possess several global minima and are invariant under a general finite reflection group G. We establish existence of nontrivial entire solutions connecting the global minima of W along certain directions at infinity.