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We prove the existence of unbounded solutions of the asymmetric oscillation in the case when each zero of the discriminative function is degenerate. This is the only case that has not been studied in the literature.

In this paper, we are concerned with unbounded solutions of the singularly perturbed forced pendulum equation in the presence of friction, namely

$$\varepsilon ^2u_\varepsilon ^{{\prime}{\prime}} + \sin u_\varepsilon = \varepsilon ^2\alpha (t)u_\varepsilon + \varepsilon ^2\beta (t)u_\varepsilon ^{\prime} \quad {\rm in}\;(-L,L).{\rm }$$

Using a limiting energy function, we describe the behaviour of the solutions as the parameter ε 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.

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The existence of unbounded solutions of asymmetric oscillations in the degenerate resonant case

Multi-clustered solutions for a singularly perturbed forced pendulum equation

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2 results

The existence of unbounded solutions of asymmetric oscillations in the degenerate resonant case

Multi-clustered solutions for a singularly perturbed forced pendulum equation