Mohsen Timoumi, Infinitely many fast homoclinic orbits for a class of superquadratic damped vibration systems, Vol. 2020 (2020), Article ID 8, pp. 1-16

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DOI: 10.23952/jnfa.2020.8

Received August 13, 2019; Accepted February 11, 2020; Published February 27, 2020

Abstract. In this paper, we consider 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 symmetric matrix valued function and $W(t,x)\in C^{1}(\mathbb{R} \times \mathbb{R}^{N}, \mathbb{R})$. We prove the existence of infinitely many fast homoclinic solutions for the system when $Q(t)=\int^{t}_{0}q(s)ds\rightarrow+\infty$ as $\left|t\right|\rightarrow\infty$, $L$ is neither coercive nor uniformly positive definite and $W(t,x)$ is superquadratic at infinity in the second variable but does not satisfy the well-known superquadratic growth conditions like the Ambrosetti-Rabinowitz or the Fei’s conditions.