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<p>In this paper, we study the existence and multiplicity of homoclinic solutions for a class of second-order Hamiltonian system: <i>u</i>″(<i>t</i>) − <i>L</i>(<i>t</i>)<i>u</i>(<i>t</i>) + ⊽<i>V</i>(<i>t,u</i>) = 0, where <i>L</i>(<i>t</i>) and <i>V</i>(<i>t,u</i>) are not periodic in <i>t</i>. First, we introduce the definition of index and establish the corresponding index theory. Then, by using the index theory and critical point theory, we prove our main results under the asymptotic quadratic conditions of the potential function.</p>

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Homoclinic Solutions for a Class of Asymptotically Quadratic Second-Order Hamiltonian System

  • 期刊论文

  • 发表日期：

    2025年8月15日

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  • Tiefeng Ye,
  • Huixing Zhang,
  • Wenbin Liu

摘要

In this paper, we study the existence and multiplicity of homoclinic solutions for a class of second-order Hamiltonian system: u″(t) − L(t)u(t) + ⊽V(t,u) = 0, where L(t) and V(t,u) are not periodic in t. First, we introduce the definition of index and establish the corresponding index theory. Then, by using the index theory and critical point theory, we prove our main results under the asymptotic quadratic conditions of the potential function.