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Subharmonic solutions for a class of Lagrangian systems

  • * Corresponding author: Marek Izydorek

    * Corresponding author: Marek Izydorek 

M. Izydorek and J. Janczewska are supported by Grant BEETHOVEN2 of the National Science Centre, Poland, no. 2016/23/G/ST1/04081

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  • We prove that second order Hamiltonian systems $ -\ddot{u} = V_{u}(t,u) $ with a potential $ V\colon \mathbb{R} \times \mathbb{R} ^N\to \mathbb{R} $ of class $ C^1 $, 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 $ 0\in \mathbb{R} ^N $. We will also discuss when subharmonics pass to a nontrivial homoclinic orbit.

    Mathematics Subject Classification: Primary: 37J45, 70H03; Secondary: 34C25, 34C37.


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