A singularly perturbed elliptic partial differential equation with an almost periodic term

was studied, where h is an almost periodic (defined in a moment) function, and F : Rn → R a “superquadratic” potential. That is, F (q) behaves like q to a power greater than 2, with F (q)/|q|2 → 0 as |q| → 0 and F (q)/|q|2 → ∞ as |q| → ∞. For example, F (q) = |q|p−1q with p > 1 would qualify. The authors found that (1.0) must have a nonzero solution homoclinic to zero. Since this result, many papers (see [CMN], [R1], and [ACM], for example) have been written concerning Hamiltonian systems with almost periodic terms. As we will see, it is natural to extend the definition of almost periodic to functions on Rn, n > 1, or even to more general topological groups. Thus one can write a PDE version of (1.0),