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.