近年来，图上的偏微分方程引起了众多学者的关注。一方面，它具有重要的理论意义，该类方程除了保持许多经典偏微分方程的良好性质之外，还具有一些特有的新的性质。关于图上的偏微分方程解的存在性、渐进性质、热核估计，以及具有几何背景的重要不等式等已有大量的研究成果。另一方面，它还具有明确的应用价值，图像处理、数据挖掘、神经网络等领域的许多问题都与此相关。 本文在变分法的框架下研究与图上的非线性偏微分方程有关的几个基本问题。我们主要关注比有限图更一般也更困难的局部有限图上的方程。整理并研究了一套与此相关的基本概念和必备工具，最终证明了相关方程的解的存在性，还进一步分析了解的渐进性质等。 在第一章介绍了相关的问题背景和研究现状之后，我们在第二章给出局部有限图 $G=(V,E)$ 上的几个基本概念及相关的基本性质。特别地，定义了局部有限图上的Sobolev空间，并证明了Sobolev 空间的自反性和完备性。为了后文的分析工作所需，我们还证明了图上的Green公式。 第三章，我们考虑如下二阶的非线性$p$-Laplacian方程\begin{equation}\label{abstract1} -\Delta_{p} u(x) +(\lambda a(x)+1)|u|^{p-2}(x)u(x)= f(x,u(x)), \ \ x\in V,\end{equation}其中$\Delta_{p}$ 为图上离散 $p$-Laplace 算子，$\lambda>1$ 且 $p\geq 2$，$a(x)\geq 0$ 为定义在 $V$ 上的函数，$f$ 为非线性函数。通过验证该方程满足山路引理的条件，我们得到了其正解的存在性。利用Nehari 流形方法，我们证明了该方程存在基态解，且证明了当 $\lambda\rightarrow\infty$ 时，其基态解收敛到对应的极限方程的基态解。 第四章，我们将上述结果推广到更困难的高阶方程，具体地，我们研究了下述双调和非线性方程\begin{equation}\label{abstract2}\Delta^{2} u(x) -\Delta u(x)+(\lambda a(x)+1)u(x)= |u(x)|^{p-2}u(x),\ \ \ \ x\in V,\end{equation}其中 $\Delta^{2}$ 为图上离散双调和算子，$\lambda>1$ 且 $p>2$，$a(x)\geq 0$ 为定义在 $V$ 上的函数。我们证明了它存在基态解，并且证明了当 $\lambda\rightarrow\infty$ 时，该方程的基态解收敛到对应的极限方程的基态解。 最后，我们在第五章对研究工作进行总结，并梳理了在未来工作中值得考虑的几个重要和有趣的问题。

Recently, partial differential equations on the graphs have attracted the attention of many mathematicians. On one hand, it has important theoretical significance.It not only has the properties of the classical partial differential equations, but also has some special and new properties.There are many works about the existence and asymptotic properties of the solutions of the partial differential equations on graphs, the estimates for heat kernel on graphs and so on. On the other hand, there are many applications in other fields, such as imagine processing, data analysis, neural network, etc. In this paper, using the variational method some basic problems about the nonlinear partial differential equations on graphs are studied. We focus on equations on the locally finite graphs that are more difficult than equations on finite graphs. In first chapter, we introduce the background of this problem. In second chapter, we give some basic concepts and introduce some basic properties of the locally finite graphs. In particular, we define Sobolev space on the locally finite graph and prove the reflexivity and completeness of Sobolev space. We also prove the Green formulas on the locally finite graphs. In chapter 3, we consider the following nonlinear $p$-Laplacian equation$$-\Delta_{p} u(x) +(\lambda a(x)+1)|u|^{p-2}(x)u(x)= f(x,u(x)), \ \ \hbox{in}\ \ V,\eqno(1)$$where $\Delta_{p}$ is the discrete $p$-Laplacian on graphs, $\lambda>1$ and $p\geq 2$ are constants, $a(x)\geq 0$ is a function defined on graphs and $f$ is the nonlinear function. We can prove that the equation has a positive solution by the Mountain Pass theorem and a ground state solution via the method of Nehari manifold, for any $\lambda>1$. In addition, as $\lambda\rightarrow +\infty$, we prove that the ground state solutions converge to a solution of the corresponding limit problem. In chapter 4, we generalize the above results to the higher order equations. Specifically, we study the following nonlinear biharmonic equations$$\Delta^{2} u -\Delta u+(\lambda a+1)u= |u|^{p-2}u,\eqno(2)$$where $\Delta^{2}$ is the discrete biharmonic operator on graphs, $\lambda>1$ and $p>2$ are constants and $a(x)\geq 0$ is a function defined on graphs. We prove that the equation has a ground state solution. We also prove as $\lambda\rightarrow +\infty$, the solutions converge to a solution of corresponding limit problem. In chapter 5, we list several important and interesting problems which will be worth considering in the future work.