Some Results On The Indefinite Kirchhoff Type Elliptic Boundary Problems

This paper mainly by using the theories of bifurcation and the methods of constrained critical points,and combining some techniques in analysis,deal with the Kirchhoff type problems involving sign-changing weight functions.In this paper we will give some results on the local and global bifurcation structure for the Kirchhoff type problems,also we will give some results on the existence and multiplicity of the indefinite Kirchhoff type problems.This paper be divided into three chapters:The first chapter will give the background of the paper.Also,some basic theories on bifurcation and the variational methods will be given in this chapter.The second chapter will deal with the following indefinite Kirchhoff type problem(?)where ? is a bounded domain of RNwith a smooth boundary,N = 2,3,W(x),V(x)?L?(?),a(x),b(x)? C?(?),? ?(0,1),a(x)? a0> 0,a0 is a constant,b(x)? 0.Under suitable conditions,by employing the re-scaling methods,Liouville theorem,Crandall-Rabinowitz local bifurcation Theorem,Rabinowitz global bifurcation Theorem,and the properties of the principal eigenvalues,we consider the existence of bifurcation points,the local and global bifurcation structure in three cases: 1)V ? 0;2)V has the same sign and V? 0;3)V changes sign.The main results of this chapter extend the related results on local elliptic boundary problems to the non-local Kirchhoff type problems.Moreover,we extend the methods obtaining a priori bounds for elliptic equation to the non-local Kirchhoff type elliptic bounary problems.The third chapter deal with the existence and multiplicity of positive solutions of the indefinite Kirchhoff type problem in the case of a = b = 1.Under suitable conditions,by using some variational methods,mainly the Nehari manifolds methods and Ekeland variational principle,we obtain some results concern-ing the existence and multiplicity of positive solutions of the indefinite Kirchhoff type problem.The main results extend some related results obtained by K.J.Brown.