Asymptotic behavior of minima and mountain pass solutions for a class of Allen-Cahn models

Abstract

where G(u) = u2(1 − u)2 is a double well potential, ε > 0, and Aε(x) = 1+A(x)/ε with 0 ≤ A ∈ C1(Rn), 1−periodic in x1, · · · , xn, Ω is the support of A|[0,1]n and has a smooth boundary, and Ω ⊂ (0, 1)n. A main result of [9] is that there is an ε0 > 0 such that for any finite set T ⊂ Zn and ε ∈ (0, ε0], (1.1) has a solution, Uε,T with 0 < Uε,T < 1, Uε,T is near 1 on A T ≡ T +Ω and near 0 on BT ≡ (Zn\T )+Ω. Moreover as ε → 0, Uε,T → 1 uniformly on AT and Uε,T → 0 uniformly on BT . When T is finite, Uε,T is characterized as the minimizer of a constrained variational problem associated with (1.1). Although Uε,T may not be unique, the set of such minimizers, Mε(T ), is ordered. The setting of [9] was further treated in [10] where it was shown that for each finite T , there is an ε1(T ) > 0 such that for ε ∈ (0, ε1(T )), (1.1) has a solution, Vε,T of mountain pass type with 0 < Vε,T < Uε,T . The main goal of this note is to study the setting of when T is finite and consists of two widely separated subsets, that is, T = T1 ∪ (l + T2) ≡ Tl for T1, T2 ⊂ Zn, l ∈ Zn and large |l| > 0. In particular we are interested in the asymptotic behavior as l → ∞ of the minimizers, Uε,Tl , and the mountain pass solutions, as well as the corresponding critical values. To describe our