An Infinite Number Of Sign Solutions For A Class Of Fourth-order Nonlinear Elliptic Boundary Value Problems

In this chapter, we consider the following problem. where ?2u denotes the biharmonic operator, let Q be a bounded open set in RN with smooth boundary, c ? K.In engineering, the fourth-order elliptic equation with the biharmonic operator ?2u+cAu= f(x, u), x?? is used to describe the deformation of an suspen-sion bridge. As the bridge is in equilibrium and there are no external forces, the corresponding equation satisfies the boundary condition u|(?)?= ?u|(?)?,=0.As to this type of fourth-order elliptic boundary value problems, Lazer and McKenna have pointed out that this type of nonlinearity furnishes a model to study travelling waves in suspension bridges in 1990. Since then more general nonlinear fourth-order elliptic boundary value problems have been studied. For problem (1.1.1) when f(x, u)= bg(x, u), Micheletti and Pistoia proved that there exist two or three solutions for a more general nonlinearity g by variational method. Zhang proved the existence of solutions for a more general nonlinearity f(x,u) under some weak assumptions. Zhang and Li proved the existence of multiple nontrivial solutions by means of Morse theory and local linking. But the existence and multiple of sign-changing solutions for (1.1.1) have not been studied except Zhou and Zhang. using the sign-changing critical theorems, Zhou got the existence of four sign-changing solutions or infinitely many sign-changing solutions for (1.1.1). Zhang used minimax method to construct sign-changing solutions to prove. In the paper, two theorems on the existence and multiplicity of the sign-changing solutions for a class of fourth-order elliptic boundary value problems are proved by using the descending flow invariant set method. The main results and the proofs are quite different from those presented by other literatures.The structure of this thesis is organized as follows:Chapter One:Introduction and Main Results. In this chapter, on one hand, we recapitulate the research background and development status of (1.1.1). on the other hand, we state the theorems about positive solutions, negative solutions, and sign-changing solutions.Chapter Two:Preliminaries. The first part, we recapitulate the related con-cepts, theorems. the second part, preparation theorems are listed which are used for proving the existence and the multiplicity of solutions in the last chapter.Chapter Three:Proof of the Main Results. Firstly, we state the corresponding functional of (1.1.1). secondly, using the descending flow invariant set method, we prove the existence of positive solutions and negative solutions of (1.1.1). thirdly, the weaker versions on the existence and multiplicity of the sign-changing solutions are proved by the descending flow invariant set method. Finally, we prove that (1.1.1)has a sign-changing solution and finitely many sign-changing solutions under weaker conditions.Chapter four:Conclusions and Outlook. Contents of the paper is summed up. we discuss if the methods in this article could be adapted to other nonlinear differential equations.