The Boundary Layer And Sign-changing Solutions For Two Types Of Differential Equations

In this thesis,we mainly study two types of differential equations,one is the singular perturbation boundary value problem,and the other is the Kirchhoff type equation.For the former we apply the theory of differential inequality to obtain the existence and asymptotic estimation of the solution.For the latter we obtain the existence of the minimum energy sign-changing solution by using constrained minimization of Nehari manifold and variational method.The thesis consists of five parts.Introduction briefly describes the research purpose and significance of this thesis.It includes research background,purpose and significance of singular perturbation equation and Kirchhoff type equation,as well as the previous research results and our main research work.In Chapter 1,we introduce the definitions,lemmas,theorems and preparatory knowledge of the two type equations.In Chapter 2,we study a singular perturbed boundary value problem.First we use synthetic expansion method to construct the formal asymptotic solution,and then apply the theory of differential inequality to obtain the existence and asymptotic estimates of the solution.In Chapter 3,we study the existence of the minimum energy sign-changing solution of the Kirchhoff type equation.Using the variational method and the constrained minimization of Nehari manifold,it is shown that there exists a minimum energy signchanging solution to the Kirchhoff type equation.In Chapter 4,we summarize the contents of this thesis,and make an outlook for the future research work.