Homoclinic orbits for damped vibration systems with asymptotically quadratic or subquadratic potentials

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• 作者：Huiwen Chen ; Zhimin He ; Zigen Ouyang ; Maoxin Liao
• 关键词：damped vibration systems ; homoclinic orbits ; variational methods ; asymptotically quadratic theorem ; subquadratic
• 刊名：Advances in Difference Equations
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：2016
• 期：1
• 全文大小：1,618 KB
• 刊物主题：Difference and Functional Equations; Mathematics, general; Analysis; Functional Analysis; Ordinary Differential Equations; Partial Differential Equations;
• 出版者：Springer International Publishing
• ISSN：1687-1847

文摘

In this paper, we study a class of damped vibration systems, $$ \ddot{u}(t)+B\dot{u}(t)-L(t)u(t)+\nabla W\bigl(t,u(t)\bigr)=0, \quad \forall t \in \mathbb{R}, $$ where \(W(t,u)\) is of indefinite sign. By using a critical point theorem of Ding, we establish a new criterion to guarantee that the above system has infinitely many nontrivial homoclinic orbits under the assumption that \(W(t,u)\) is asymptotically quadratic or subquadratic as \(|u|\rightarrow\infty\). Recent results in the literature are generalized and significantly improved. Keywords damped vibration systems homoclinic orbits variational methods asymptotically quadratic theorem subquadratic MSC 34C37 35A15 37J45 1 IntroductionIn this paper, we consider the following damped vibration system: $$ \ddot{u}(t)+B\dot{u}(t)-L(t)u(t)+\nabla W\bigl(t,u(t)\bigr)=0, \quad \forall t\in \mathbb{R}, $$ (1.1) where \(u\in\mathbb{R}^{N}\), B is an antisymmetric \(N\times N\) constant matrix, \(L\in C(\mathbb{R},\mathbb{R}^{N\times N})\) is a symmetric matrix-valued function, and \(W\in C^{1}(\mathbb{R}\times \mathbb{R}^{N},\mathbb{R})\). As usual, we say that a solution u of system (1.1) is homoclinic to zero if \(u\in C^{2}(\mathbb {R},\mathbb{R}^{N})\), \(u(t)\rightarrow0\), and \(\dot{u}(t)\rightarrow 0\) as \(|t|\rightarrow\infty\). In addition, if \(u(t)\not\equiv0\), then \(u(t)\) is called a nontrivial homoclinic solution.Homoclinic orbits have been found in various models of continuous dynamical systems and play an important role in the study of the behavior of dynamical systems; see [1]. Thus, the study of homoclinic orbits has become one of the most important directions in the research of dynamical systems.