Infinitely many fast homoclinic solutions for damped vibration systems with locally defined potentials

Abstract

In this paper, we prove the existence of infinitely many fast homoclinic solutions for the nonperiodic damped vibration system $$\ddot{u}(t)+(q(t)I_{N}+B)\dot{u}(t)+\frac{1}{2}q(t)Bu(t)-L(t)u(t)+\nabla W(t,u(t))=0,\ \forall t\in\mathbb{R} \leqno(1)$$
where $q\in C(\mathbb{R},\mathbb{R})$, $I_{N}$ is the $N\times N$ identity matrix, $B$ is an antisymmetric $N\times N$ constant matrix, $L\in C(\mathbb{R},\mathbb{R}^{N^{2}})$ is a symmetric matrix valued function unnecessary coercive and $W(t,x)\in C^{1}(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R})$ is only locally defined and superquadratic near the origin with respect to $x$. Our results extend and improve some existing results in the literature.