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Infinitely many homoclinic solutions for damped vibration systems with locally defined potentials
  • Wafa Selmi,
  • Mohsen Timoumi
Wafa Selmi
University of Monastir Faculty of Sciences of Monastir

Corresponding Author:[email protected]

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Mohsen Timoumi
University of Monastir Faculty of Sciences of Monastir
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Abstract

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.