The Existence Of Solutions For Four-order Differential Equations

From the beginning of 20th Century, the boundary value problem of differential equations has gradually become a hot issue in the research of the differential equations, especially the existence of solutions or multiple solutions of Dirichlet and Neumann boundary value problem. Over the years, due to the extensive application of the differential equation boundary value problems in the fields of physics, aerospace and biology,many linear and quasi linear elliptic equation boundary value problems have been gotten relatively abundant results. Under this background, the boundary value problems of the double harmonic equation and p-harmonic equation have attracted more attention. However, since the p(x)-biharmonic operator has relatively complex nonlinear property,many classical theories and methods cannot be used, and the research content of the Neumann boundary value problems involving p(x)-biharmonic operator is relatively limited. Therefore, it is of great practical significance to study the boundary value problem of this kind of differential equation.In this paper, by using the variational method and the different critical points theorems, we study the existence of solutions and the number of solutions for Neumann boundary value problems of p(x)-biharmonic operator with continuous or discontinuous nonlinear term respectively, and get some new results.In chapter one, we introduce research background, significance and method of the Neumann boundary value problems involving p(x)-biharmonic operator. Besides, we review the research background and status of the four order nonlinear differential equation boundary value problem. Finally, the main research contents of this paper are described.In chapter two, we introduce critical point theorems involved in the research of this paper.In chapter three, by using the variational method and Ricceri three critical points theorem, we study the problem of continuous nonlinear Neumann boundary value problem with p(x)-biharmonic operator, and get the result of the existence of at least three solutions.In chapter four, by using the variational method and nonsmooth critical point theorem, we study the problem of discontinuous nonlinear Neumann boundary value problems with p(x)-biharmonic operator, and get the result of the existence of one solution.In chapter five, the research contents and main research results are summarized, and the future research work is prospected.