文章基本信息

标题：Homoclinic orbits for a class of second order dynamic equations on time scales via variational methods
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作者：You-Hui Su ; Xingjie Yan ; Daihong Jiang 等
期刊名称：Advances in Difference Equations
印刷版ISSN：1687-1839
电子版ISSN：1687-1847
出版年度：2017
卷号：2017
期号：1
页码：47
DOI：10.1186/s13662-017-1098-1
语种：English
出版社：Hindawi Publishing Corporation
摘要：In this paper, we study the existence of nontrivial homoclinic orbits of a dynamic equation on time scales T $\mathbb{T}$ of the form { ( p ( t ) u Δ ( t ) ) Δ + q σ ( t ) u σ ( t ) = f ( σ ( t ) , u σ ( t ) ) , △ -a.e. t ∈ T , u ( ± ∞ ) = u Δ ( ± ∞ ) = 0 . $$ \left \{ \textstyle\begin{array}{l} ( p(t)u^{\Delta}(t) ) ^{\Delta}+q^{\sigma}(t)u^{\sigma}(t)= f(\sigma(t),u^{\sigma}(t)),\quad \triangle\text{-a.e. } t\in\mathbb{T}, \\ u(\pm\infty)=u^{\Delta}(\pm\infty)=0. \end{array}\displaystyle \right . $$ We construct a variational framework of the above-mentioned problem, and some new results on the existence of a homoclinic orbit or an unbounded sequence of homoclinic orbits are obtained by using the mountain pass lemma and the symmetric mountain pass lemma, respectively. The interesting thing is that the variational method and the critical point theory are used in this paper. It is notable that in our study any periodicity assumptions on p ( t ) $p(t)$ , q ( t ) $q(t)$ and f ( t , u ) $f(t,u)$ are not required.
关键词：time scales ; variational structure ; homoclinic orbits ; critical point theorem