Multiplicity of Entire Solutions For a Class of Almost Periodic Allen-Cahn Type Equations

Abstract

Abstract We consider a class of semilinear elliptic equations of the form −Δu(x,y) + a(εx)Wʹ(u(x,y)) = 0, (x,y) ∊ ℝ2 (0.1) where ε > 0, a : ℝ → ℝ is an almost periodic, positive function and W : ℝ → ℝ is modeled on the classical two well Ginzburg-Landau potential W(s) = (s2 - 1)2. We show via variational methods that if ε is sufficiently small and a is not constant then (0.1) admits infinitely many two dimensional entire solutions verifying the asymptotic conditions u(x, y) → ±1 as x → ±∞ uniformly with respect to y ∊ ℝ.