几类格点动力系统的变分问题研究

We mainly study the existence of periodic solutions and homoclinic orbits of lattices dynamical systems with nearest neighbor interaction potentials and the existence and multiplicity of homoclinic orbits for some discrete nonlinear Schrödinger equations in one dimensional lattices by using variational methods, enrich and consummate the well known results. Our main contents are as follows: 1) we study the existence and multiplicity of homoclinic orbits of discrete nonlinear Schrödinger equations with unbounded potentials and more general nonlinearities by using the variational tools; 2) we study the existence of ground state homoclinic orbits for some discrete nonlinear Schrodinger equations with asymptotic periodic and bounded non-periodic potentials by using the representation of Palais-Smale sequences and the concentration-compactness principle and so on, respectively; 3) we study the existence and multiplicity of travelling waves for some lattices dynamical systems with more general on-site and nearest neighbor interaction potentials by using the minimax methods and so on; 4) we study the existence and multiplicity of periodic solutions for the FPU types lattices dyanmical systems with asymptotic superquadratic growth potentials by using the spectrum theory and periodic approximations techniques in combination with varational tools. By proposing and solving the problems in the project, it will help to deepen studies of the lattice dynamical systems, and it will extend applications of this theory in related fields.

本项目主要利用变分工具研究一维格点上带有相邻相互作用势的格点动力系统的周期解和行波解以及离散非线性薛定谔方程的同宿轨的存在性和多重性，丰富和完善已有结果。主要内容包括：1）利用变分工具研究带有更一般非线性项时无界位势情形离散非线性薛定谔方程的同宿轨的存在性和多重性；2)利用Palais-Smale序列分解技巧以及集中紧性原理等工具分别研究渐近周期位势情形离散非线性薛定谔方程以及一类有界非周期位势情形离散非线性薛定谔方程的基态同宿轨的存在性；3）利用极小极大方法等工具研究带有更一般格点势和相邻相互作用势的格点动力系统的行波解的存在性和多重性；4）利用谱理论、周期逼近技巧及变分工具等研究渐近二次位势情形时FPU型格点动力系统周期运动的存在性和多重性。本项目中问题的提出和解决，将有助于深化格点动力系统的研究并拓展该理论在相关领域中的应用。

非线性格点动力系统是非常重要的固有离散模型，具有非常强的物理背景。本项目主要利用变分工具研究一维格点上由无穷多个带有相邻相互作用势的粒子构成的FPU型格点动力系统的行波解以及一维格点上离散非线性薛定谔方程同宿轨的存在性和多重性问题，另外我们还研究了Kirchhoff型方程、Schrödinger-Bopp-Podolsky系统以及Klein-Gordon-Maxwell系统等几类典型非局部椭圆方程问题的解的存在性与多重性及解的性态等问题。主要结果包括：1) 得到了具有超二次增长的势函数的FPU系统行波解的存在性和单调性结果；2）得到了带有周期位势和局部超二次增长条件的离散非线性薛定谔方程的基态解以及无穷多个结构不同的解的存在性结果；3）得到了一类带有非周期位势的离散非线性薛定谔方程的基态解的存在性和非存在性结果；4）得到了带有强制位势的基尔霍夫型方程和分数阶Klein-Gordon-Maxwell系统的无穷多个变号解的存在性结果；5）得到了一类带有次临界或者临界非线性项的Schrödinger-Bopp-Podolsky系统的基态解的存在性结果。本项目中问题的提出和解决，将有助于深化格点系统以及非局部椭圆方程问题的研究并拓展这些理论在相关领域中的应用。