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 Infinitely many homoclinic solutions for damped vibration systems with locally defined potentials Commun. Korean Math. Soc.Published online May 13, 2022 Wafa Selmi and Mohsen Timoumi Faculty of Sciences of Monastir; Faculty of Sciences of Monastir 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. Keywords : Damped vibration systems; homoclinic solutions; variational methods; locally defined potentials, symmetric mountain pass theorem MSC numbers : 34C37, 35A15, 37J45, 49J40 Full-Text :

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