Unbounded solutions for a periodic phase transition model

In an earlier paper, [1], the authors treated a family of Allen–Cahn model problems for which 0 and 1 are solutions and further solutions were found that are near 1 on a prescribed set, T+ΩT+Ω, where T⊂ZnT⊂Zn, and near 0 on (Zn∖T)+Ω(Zn∖T)+Ω. Here Ω⊂(0,1)nΩ⊂(0,1)n. In this paper, a more general class of potentials is treated for which the pair, {0,1}{0,1}, is replaced by ZZ and the existence of a far richer structure of shadowing solutions, including unbounded ones, is established.