Infinitely Many Homoclinic Solutions for a Class of Indefinite Perturbed Second-Order Hamiltonian Systems

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• 作者：Liang Zhang ; Xianhua Tang ; Yi Chen
• 关键词：Bolle’s perturbation method ; broken symmetry ; perturbed Hamiltonian system ; homoclinic solutions
• 刊名：Mediterranean Journal of Mathematics
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：13
• 期：5
• 页码：3673-3690
• 全文大小：632 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Birkh盲user Basel
• ISSN：1660-5454
• 卷排序：13

文摘

In this paper, we study the existence of infinitely many homoclinic solutions of the perturbed second-order Hamiltonian system$$-\ddot{u}(t)+L(t)u=W_u(t,u(t))+G_u(t,u(t)),$$where \({L(t)}\) and \({W(t,u)}\) are neither autonomous nor periodic in \({t}\). Under the assumptions that \({W(t,u)}\) is indefinite in sign and only locally superquadratic as \({|u|\to +\infty}\) and \({G(t,u)}\) is not even in \({u}\), we prove the existence of infinitely many homoclinic solutions in spite of the lack of the symmetry of this problem by Bolle’s perturbation method in critical point theory. Our results generalize some known results and are even new in the symmetric case.KeywordsBolle’s perturbation methodbroken symmetryperturbed Hamiltonian systemhomoclinic solutionsThis work is partially supported by the NNSF (No. 11571370) of China, Natural Science Foundation of Jiangsu Province of China (No. BK20140176) and the NSF of Shandong Province of China (No. ZR2014AP011).