# Subharmonic and homoclinic solutions for a class of Hamiltonian Systems

### Abstract

We study the existence of infinitely many subharmonic solutions for a class of dynamical systems

$$ \ddot{u}(t)+A\dot{u}(t)+V'(t,u(t))=0,$$ where A is a skew-symmetric constant matrix, t $\in\mathbb{R}$, $u\in\mathbb{R}^N$ and V $\in C^1(\mathbb{R}\times\mathbb{R}^N,\mathbb{R})$,

$V(t,u)=-K(t,u)+W(t,u)$ is T-periodic with respect to t, $T>0$. We assume that $K$ satisfies the pinching condition and $W$ satisfies a new superquadratic condition instead of the Ambrosetti-Rabinowitz condition. Furthermore, we can get the existence of homoclinic solutions by taking the limit of subharmonics. This results

generalize and improve some existing findings in the known literature.

Published

2018-08-25

Section

Articles