Entire solutions in ${\mathbb{R}}^{2}$ for a class of Allen-Cahn equations

Abstract

We consider a class of semilinear elliptic equations of the form 15.7cm - where , is a periodic, positive function and is modeled on the classical two well Ginzburg-Landau potential . We look for solutions to ([see full textsee full text]) which verify the asymptotic conditions as uniformly with respect to . We show via variational methods that if e is sufficiently small and a is not constant, then ([see full textsee full text]) admits infinitely many of such solutions, distinct up to translations, which do not exhibit one dimensional symmetries.