Layer structures for the solutions to the perturbed simple pendulum problems

We consider the perturbed simple pendulum equation −u (t) = μf u(t) + λ sin u(t), t ∈ I := (−T,T ), u(t) > 0, t∈ I, u(±T ) = 0, where λ > 0 and μ ∈ R are parameters. The typical example of f is f (u) = |u|p−1u (p>1). The purpose of this paper is to study the shape of the solutions when λ 1. More precisely, by using a variational approach, we show that there exist two types of solutions: one is almost flat inside I and another is like a step function with two steps.  2005 Elsevier Inc. All rights reserved