Multi-clustered solutions for a singularly perturbed forced pendulum equation

Abstract

In this paper, we are concerned with unbounded solutions of the singularly perturbed forced pendulum equation in the presence of friction, namely e(2)u(x)" e + sinue = e2a( t)ue + e2 beta(t)u e in (- L, L). Using a limiting energy function, we describe the behaviour of the solutions as the parameter e approaches zero. We also prove the existence of a family of solutions having a prescribed asymptotic profile and exhibiting a highly rotatory behaviour alternated with a highly oscillatory behaviour in some open subsets of the domain. The proof relies on a combination of the Nehari finite dimensional reduction with the topological degree theory.