Hamilton型椭圆方程组是偏微分方程研究的重要领域之一, 它在物理学、工程学、经济学等多个领域具有重要的应用背景. 例如, 在物理学中, Hamilton型方程组可以描述一个物理系统的动力学行为, 揭示广义坐标、广义动量、Hamilton量等物理量之间的内在联系. 在经济学中, Hamilton函数被引入动态优化问题之中, 作为宏观经济分析的一个重要手段. 因此, 对于Hamilton型方程组的研究具有重要的理论和实际意义.本文主要研究Hamilton型椭圆方程组周期解的存在性、多解性和非存在性等. 具体而言, 我们将考虑如下方程组:\begin{equation*}\left\{\begin{array}{ll}-\Delta u=H_v(y, u, v),\;\;\; &y\in\mathbb R^N, \\-\Delta v=H_u(y, u, v),\;\;\; &y\in\mathbb R^N.\end{array}\right.\end{equation*}主要结果有以下三个方面:首先, 注意到如果Hamilton函数$H(y, u, v)=\frac{1}{2}v^2+\frac{N-4}{2N}K(y)u^{\frac{2N}{N-4}}$时, 方程组退化为方程$(-\Delta)^2 u = K(y) u^{\frac{N+4}{N-4}}.$ 更一般地, 我们研究高阶预定曲率方程的周期解. 利用Lyapunov-Schmidt约化, 在$K(y)$的周期性条件下我们构造了高阶预定曲率方程的周期解.其次, 我们研究$H(y,u,v)=\frac{1}{p+1} v^{p+1} + \frac{1}{q+1}K(y)u^{q+1}$时的临界Hamilton型方程组（I型方程组）. 通过对极限方程组基态解在$p< \frac{N}{N-2}$时渐近行为的分析, 选取适当的近似解, 进行能量估计后得到了关于前$k$个分量周期的解. 除此之外, 我们还证明了当$k$在一定范围时周期解的不存在性.最后, 我们研究更一般$H(y,u,v)=\frac{1}{p+1}K(y)v^{p+1} + \frac{1}{q+1}K(y)u^{q+1}$时的临界Hamilton型方程组（II型方程组）. 在$p\geq \frac{N}{N-2}$的情形下利用Green函数对近似解进行估计, 通过约化方法构造得到条带状区域上的单泡泡解, 并周期延拓到$\mathbb R^N$后最终得到全空间上的周期解.
Hamiltonian elliptic systems are one of the important fields in partial differential equations, with significant applications in physics, engineering, economics, and other fields. For instance, in physics, Hamiltonian systems can describe the dynamic behavior of a physical system, revealing the relationships among generalized coordinates, generalized momenta, and Hamiltonian quantities. In economics, Hamiltonian function is introduced into dynamic optimization problems as a tool for macroeconomic analysis. Therefore, the study of Hamiltonian systems has theoretical and practical significance.This paper studies the existence, multiplicity and non-existence of periodic solutions for Hamiltonian elliptic systems. Specifically, we consider the following system:\begin{equation*}\left\{\begin{array}{ll}-\Delta u=H_v(y, u, v),\;\;\; &y\in\mathbb R^N, \\-\Delta v=H_u(y, u, v),\;\;\; &y\in\mathbb R^N.\end{array}\right.\end{equation*}The main results are in the following three aspects:Firstly, we note that if Hamiltonian function $H(y, u, v)=\frac{1}{2}v^2+\frac{N-4}{2N}K(y)u^{\frac{2N}{N-4}}$, then the system becomes the equation $(-\Delta)^2 u = K(y) u^{\frac{N+4}{N-4}}.$ More generally, we study the periodic solutions of the higher order prescribed curvature equation. Using the Lyapunov-Schmidt reduction, we construct periodic solutions for this equation under the periodic condition of $K(y)$.Secondly, we study the critical Hamiltonian system when $H(y,u,v)=\frac{1}{p+1} v^{p+1} + \frac{1}{q+1}K(y)u^{q+1}$(system I). By analyzing the asymptotic behavior of the ground state solution for the limit system when $p< \frac{N}{N-2}$, we select approximate solutions and use energy estimates to obtain solutions which are periodic in first $k$ variables. In addition, we prove the non-existence of periodic solutions when $k$ is within a certain range.Lastly, we study the critical Hamiltonian system for the more general case $H(y,u,v)=\frac{1}{p+1}K(y)v^{p+1} + \frac{1}{q+1}K(y)u^{q+1}$(system II). Under the condition $p\geq \frac{N}{N-2}$, we use the Green's function to estimate the approximate solutions, construct single bubble solutions in a strip through reduction methods, and finally obtain periodic solutions in the entire space after periodic extension to $\mathbb R^N$.
