Existence of weak quasi-periodic solutions for a second order Hamiltonian system with damped term via a PDE approach

Abstract

In this paper, we investigate the existence of weak quasi-periodic solutions for the second order Hamiltonian system with damped term: \begin{equation} \ddot{u}(t)+q(t)\dot{u}(t)+D W(u(t))=0, \qquad t\in \mathbb R,\tag{HSD} \end{equation} where $u:\mathbb R\to \mathbb R^n$, $q:\mathbb R\to \mathbb R$ is a quasi-periodic function, $W:\mathbb R^n\to \mathbb R$ is continuously differentiable, $DW$ denotes the gradient of $W$, $W(x)=-K(x)+F(x)+H(x)$ for all $x\in \mathbb R^n$ and $W$ is concave and satisfies the Lipschitz condition. Under some reasonable assumptions on $q, K,F,H$, we obtain that system has at least one weak quasi-periodic solution. Motivated by Berger et al. (1995) and Blot (2009), we transform the problem of seeking a weak quasi-periodic solution of system (HSD) into a problem of seeking a weak solution of some partial differential system. We construct the variational functional which corresponds to the partial differential system and then by using the least action principle, we obtain the partial differential system has at least one weak solution. Moreover, we present two propositions which are related to the working space and the variational functional, respectively.

Keywords

• weak quasi-periodic solution
• second order hamiltonian system
• damped term
• variational method
• pde approach