Partial Differential Equations On The Graphs

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-?_pu(x)+(?a(x)+1)|u|^p-2(x)u(x)=f(x,u(x)),in V,(1)where?_pis the discrete p-Laplacian on graphs,?>1 and p?2 are constants,a(x)?0is 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?>1.In addition,as??+?,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.Specifi-cally,we study the following nonlinear biharmonic equations?²u-?u+(?a+1)u=|u|^p-2u,(2)where?²is the discrete biharmonic operator on graphs,?>1 and p>2 are constants and a(x)?0 is a function defined on graphs.We prove that the equation has a ground state solution.We also prove as??+?,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.