Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth

Alessandro Fonda,Rodica Toader

doi:10.1515/anona-2017-0040

Alessandro Fonda, Rodica Toader

Open Access

https://doi.org/10.1515/anona-2017-0040

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Abstract

AbstractWe prove the existence and multiplicity of subharmonic solutions for Hamiltonian systems obtained as perturbations ofNplanar uncoupled systems which, e.g., model some type of asymmetric oscillators. The nonlinearities are assumed to satisfy Landesman–Lazer conditions at the zero eigenvalue, and to have some kind of sublinear behavior at infinity. The proof is carried out by the use of a generalized version of the Poincaré–Birkhoff Theorem. Different situations, including Lotka–Volterra systems, or systems with singularities, are also illustrated.

Highlights

Introduction and main resultWe are interested in finding periodic solutions of a nonautonomous Hamiltonian system in R2N
We prove the existence and multiplicity of subharmonic solutions for Hamiltonian systems obtained as perturbations of N planar uncoupled systems which, e.g., model some type of asymmetric oscillators
In assumption (A2) we have the well-known Landesman–Lazer conditions: they will force the large-amplitude solutions of the uncoupled planar systems to rotate around the origin

Summary

Introduction and main result

We are interested in finding periodic solutions of a nonautonomous Hamiltonian system in R2N. In assumption (A2) we have the well-known Landesman–Lazer conditions: they will force the large-amplitude solutions of the uncoupled planar systems to rotate around the origin. This property, which might have an independent interest, has already been exploited in [3, 4, 9, 23], and is stated in Lemma 2.5 below. We will find a T-periodic solution of (1.1) with ε = 0, which will be used in a change of variables, in order to have the origin as a constant solution This will enable us to compute the rotation number on each planar subsystem, so to apply a generalized version of the Poincaré–Birkhoff Theorem recently obtained in [14]. We have proved that system (2.2) has a T-periodic solution

A change of variables

The rotational lemma

End of the proof

Proof of the corollaries and final remarks