A Singularly Perturbed Elliptic Partial Differential Equation with an Almost Periodic Term Gregory S. Spradlin United States Military Academy West Point, New York, USA 1. Introduction In [STT], a Hamiltonian system of the form (1.0) −u ′′ + u = h(t)∇F (u) was studied, where h is an almost periodic (defined in a moment) function, and F : R n → R a “su- perquadratic” 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−1 q 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 R n , n> 1, or even to more general topological groups. Thus one can write a PDE version of (1.0), (1.1) −Δu + u = h(x)f (u), wherein h is almost periodic and the primitive F of f satisfies appropriate superquadraticity and growth conditions. Then one may ask, does (1.1) have a “homoclinic-type” solution? That is, is there a nonzero solution u with |u(x)| + |∇u(x)|→ 0 as |x|→∞? Here we take a step towards answering in the affirmative. Let us define an almost periodic function on R n (R is a special case, and defining an a.p. function on other topological groups is an obvious generalization). First, a set A⊂ R n is relatively dense if there exists L> 0 such that for every x ∈ R n , there exists y ∈A with |x − y| <L. Next, for ǫ> 0, v ∈ R n , and h : R n → R, we say v is an ǫ-almost period of h if for all x ∈ R n , |h(x + v) − h(x)| <ǫ. Finally, h is defined to be almost periodic if for every ǫ> 0, there exists a relatively dense set A≡A(ǫ) ⊂ R n such that for all a ∈A, a is an ǫ-almost period of h. For properties of almost periodic functions (many properties of a.p. functions on R extend to a.p. functions on R n ), see [Be], [Bo], [C], [Z]. We will look at an equation similar to (1.1), of the form (1.2) −ǫ 2 Δ˜ u + V (x)˜ u = f (˜ u) on R n . Equations like (1.2) arise in the study of the nonlinear Schr¨ odinger equation and have been the subject of much study recently (see [R2], [FdP1-3], [Li] and the references therein). We will assume that V and f satisfy the following conditions: 1