Layer structures for the solutions to the perturbed simple pendulum problems

We consider the perturbed simple pendulum equation−u″(t)=μf(u(t))+λsinu(t),t∈I:=(−T,T),u(t)>0,t∈I,u(±T)=0, where λ>0λ>0 and μ∈Rμ∈R are parameters. The typical example of f   is f(u)=|u|p−1uf(u)=|u|p−1u(p>1)(p>1). The purpose of this paper is to study the shape of the solutions when λ≫1λ≫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.