A new proof for the existence of an equivariant entire solution connecting the minima of the potential for the system {\Delta}u - W_u (u) = 0

TL;DR

This paper presents a new proof for the existence of equivariant entire solutions connecting minima in a potential system, avoiding pointwise constraints in the minimization process.

Contribution

It introduces a novel proof method for the existence of solutions in a symmetric potential system, bypassing previous pointwise constraint techniques.

Findings

Existence of equivariant solutions connecting minima confirmed.

New proof method simplifies the previous approach.

Solutions exhibit specific symmetry and asymptotic behavior.

Abstract

Recently, Giorgio Fusco and the author studied the system {\Delta}u - W_u (u) = 0 for a class of potentials that possess several global minima and are invariant under a general finite reflection group, and established existence of equivariant solutions connecting the minima in certain directions at infinity, together with an estimate. In this paper a new proof is given which, in particular, avoids the introduction of a pointwise constraint in the minimization process.