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
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