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

  • Mohsen Timoumi
  • A. Raouf Chouikha


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.