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

In this paper, we are concerned with the existence of infinitely many fast homoclinic solutions for the following damped vibration system $$\ddot{u}(t)+q(t)\dot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0,\ \forall t\in\mathbb{R} $$ where $q\in C(\mathbb{R},\mathbb{R})$, $L\in C(\mathbb{R},\mathbb{R}^{N^{2}})$ is a symmatric and positive definite matix-valued function and $W\in C^{1}(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R})$. The novelty of this paper is that, assuming that $L$ is bounded from below unnecessarily coercive at infinity, and $W$ is only locally defined near the origin with respect to the second variable, we show that $(1)$ possesses infinitely many homoclinic solutions via a variant symmetric mountain pass theorem.