Communications of the
Korean Mathematical Society CKMS

In this paper, we are concerned with the existence of infinitely many fast homoclinic solutions for the following damped vibration system $$ddot{u}(t)+q(t)dot{u}(t)-L(t)u(t)+
abla W(t,u(t))=0, forall tinmathbb{R}, leqno(1)$$ where $qin C(mathbb{R},mathbb{R})$, $Lin C(mathbb{R},mathbb{R}^{N^{2}})$ is a symmetric and positive definite matix-valued function and $Win C^{1}(mathbb{R} imesmathbb{R}^{N},mathbb{R})$. 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.

Keywords: Damped vibration systems, homoclinic solutions, variational methods, locally defined potentials, symmetric mountain pass theorem

MSC numbers: Primary 34C37, 35A15, 37J45, 49J40