An asymptotic monotonicity formula for minimizers of elliptic systems of Allen-Cahn type and the Liouville property

Abstract

We prove an asymptotic monotonicity formula for bounded, globally minimizing solutions (in the sense of Morse) to a class of semilinear elliptic systems of the form Δu = Wu(u), x ∈ ℝn, n ≥ 2, with W : ℝ, → ℝ, m ≥ 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.