Equivariant entire solutions to elliptic systems with variational structure

We consider the elliptic system
$$ \Delta u = W_u(u), \quad
x\in\mathbb{R}^n, $$
for a class of potentials $W : \mathbb{R}^n \to\mathbb{R}$ that possess
several global minima and are
invariant under a finite or discrete reflection group $G$ acting on
$\mathbb{R}^n$. We establish existence of nontrivial $G$−equivariant
entire solutions $u : \mathbb{R}^n \to\mathbb{R}^n$ connecting the
global minima of $W$. If $G$ is a discrete (infinite) group the solution
$u$ has a kind of crystalline periodic structure and existence is ensured
provided the elementary cell contains a ball of radius $R^*$
with $R^*$ a constant that depends only on $W$.