Energy estimates for minimizers to a class of elliptic systems of Allen-Cahn type and the Liouville property

We prove a theorem for the growth of the energy of bounded, globally minimizing solutions to a class of semilinear elliptic systems of the form $\mathrm{\Delta }u=\mathrm{\nabla }W(u)$, $x\in {\mathbb{R}}^n$, $n\ge 2$, with $W:$\mathbb{R}$^m\to $\mathbb{R}$$,$m\geq 1$, nonnegative and vanishing at exactly one point (at least in the closure of the image of the considered solution $u$). As an application, we can prove a Liouville type theorem under various assumptions.