A Hamiltonian System with an Even Term

In this paper we study, using variational methods, a Hamiltonian system of the form −u′′ + u = h(t)V (u), where h and V are differentiable, h is positive, bounded, and bounded away from zero, and V is a “superquadratic” potential. That is, V behaves like q to a power greater than 2, so |V (q)| = o(|q|) for |q| small and V (q) > O(|q|) for |q| large. To prove that a solution homoclinic to zero exists, one must assume additional hypotheses on h (see [EL] for a counterexample). In [R1], solutions were found when h is assumed to be periodic. In [STT], solutions were found when h is almost periodic (a weaker condition than periodicity). In [MNT], a condition yet weaker than almost periodic is defined, and solutions to the equation are found when h satisfies that condition. Like periodicity and almost periodicity, this condition assumes basically that h is similar to translates of itself, that is, for certain large values of T , the functions t 7→ h(t) and t 7→ h(t + T ) are close to each other. Other ways to guarantee solutions involve making |h′| small: see papers such as [FW], [WZ], and [FdP] on the nonlinear Schrödinger equation, and [A] for a novel example of an h which “oscillates slowly”. In this paper we attempt to find solutions to the system without assuming that h satisfies any kind of time-recurrence property or restriction on h′. We assume two conditions: first, that h is even (h(−t) = h(t)). Therefore it is convenient to treat the system as a system on the half-line R = [0,∞). Second,