Existence and multiplicity of weak quasi-periodic solutions for second order Hamiltonian system

Abstract

In this paper we investigate the existence and multiplicity of weak quasi-periodic solutions for the second order Hamiltonian system $\frac{d[P(t)\dot{u}(t)]}{dt}+\nabla F(t,u(t))=0$ , $t\in\mathbb{R}$ , where $P(t)=(p_{ij}(t))_{N\times N}$ is a symmetric and continuous $N\times N$ matrix-value function on $\mathbb{R}$ and $F(t,x)$ is almost periodic in t uniformly for $x\in\mathbb{R}^{N}$ . When F has superquadratic growth, we see that the system has at least one nonconstant weak quasi-periodic solution and when the assumption $F(t,-x)=F(t,x)$ is also made, we see that the system has infinitely many weak quasi-periodic solutions by variational method.