The Existence Of Solutions For Foutth-Order Differential Equation With Sturm-Liouville Impulsive Boundary Value

Variational method is a powerful tool which is applied more and more on the study of boundary-value problems with impulsive effects, especially of the problems with Neumann and Dirichlet boundary conditions. Also, as more and more people began to study the problem of this kind, more and more critical point theorems emerge, which in turn enriches the theory of variational method.In this paper, we mainly study the existence of solutions and positive solutions, multiple solutions for fourth-order impulsive Sturm-Liouville boundary value differential problem by using the variational method and selecting different critical point theorems. Furthermore, we study two kinds of p(x)-Laplacian Dirichlet problem and p(x)-Laplacian Neumann problem. The full paper is divided into seven chapters:The first chapter is the introduction, in this chapter, we briefly introduce fourth-order Sturm-Liouville boundary-value problems with impulsive effects and some common methods to study them. The variational method with its historical background and research status are also introduced, followed by the main research contents of this paper.In chapter two, we introduce some basic concepts and the related critical point theorems which prepare for the later study.In chapter three, we study the application of variational method in the existence of multiple solution of the fourth-order Sturm-Liouville differential boundary-value problem with impulsive effects. By choosing several different critical point theorems, combing with the corresponding different assumptions, relevant conclusions of the existence of solutions are obtained.Under the premise of using the variational method, in the fourth chapter, we study a Dirichlet boundary value problem with the Laplace differential operator. By using Ricceri’variational principle,the existence of at least three weak solutions are obtained.In the chapter five, we study a Neumann boundary-value problem with p(x)-Laplacian operator. The existence results of infinitely many weak solutions and a weak solution are obtained. The main ideas involve variational methods.Finally in the last chapter, the research contents of this paper are summarized, and the possible follow-up research is prospected.