Subharmonic solutions for a class of Lagrangian systems

Anouar Bahrouni,Marek Izydorek,Joanna Janczewska

doi:10.3934/dcdss.2019121

Abstract

We prove that second order Hamiltonian systems \begin{document}$ -\ddot{u} = V_{u}(t,u) $\end{document} with a potential \begin{document}$ V\colon \mathbb{R} \times \mathbb{R} ^N\to \mathbb{R} $\end{document} of class \begin{document}$ C^1 $\end{document} , periodic in time and superquadratic at infinity with respect to the space variable have subharmonic solutions. Our intention is to generalise a result on subharmonics for Hamiltonian systems with a potential satisfying the global Ambrosetti-Rabinowitz condition from [ 14 ]. Indeed, we weaken the latter condition in a neighbourhood of \begin{document}$ 0\in \mathbb{R} ^N $\end{document} . We will also discuss when subharmonics pass to a nontrivial homoclinic orbit.

Highlights

A variational approach to the study of periodic solutions of Hamiltonian systems was initiated by Poincare at the end of the XIX century
In the first half of the XX century, Morse and Lusternik-Shnirelman theories significantly contributed to the development of research in this direction
In the second half of the XX century, the mountain pass theorem, Ekeland’s principle, linking theorems and Conley theory played an important role in the study of periodic orbits

Summary

Introduction

A variational approach to the study of periodic solutions of Hamiltonian systems was initiated by Poincare at the end of the XIX century. Where K, F : R × RN → R are C1-smooth mappings which are T -periodic in t (for some T > 0) and satisfy the following conditions: (V 1) there are constants b1, b2 > 0 such that for all (t, u) ∈ R × RN , b1|u|2 ≤ K(t, u) ≤ b2|u|2, (V 2) for all (t, u) ∈ R × RN , K(t, u) ≤ (Ku(t, u), u) ≤ 2K(t, u), (V 3) there exist r > 0, μ > 2 and 0 < ν < b1 such that for all t ∈ R, (i) 0 < μF (t, u) ≤ (Fu(t, u), u) if |u| ≥ r, (ii) 2F (t, u) ≤ (Fu(t, u), u) and |F (t, u)| ≤ ν|u|2 if |u| < r.

Results

Conclusion