Multiplicity Results For The Kirchhoff-type Equation

In recent years,some scholars apply wide-range variational techniques to establish the existence and the various behaviors of solutions to the Kirchhoff equation,but the non-local term brings some difficulties.Up to now,to our knowledge,there is no results on combining the lower and upper solution method with variational method to discuss the existence of solutions of Kirchhoff-type equations.This article considered elliptic problems of Kirchhoff-type for the case that the nonlinearity term is sign-changing.we will discuss the existence of solutions to the singular Kirchhoff equation by establishing a new upper and lower solution method.Secondly,we discussed with the Kirchhoff-type equations with non-homogeneous term.Due to the existence of a(x)and b(x)depend on x,there are fewer results on the existence of solutions for such kinds of problems.Using the method of sub-supersolutions and the fixed-point index method,we are concerned with the sign-changing solutions and multiplicity results of the Kirchhoff-type problem.In Chapter 1,we consider the following nonlocal elliptic problems where ? is a smooth bounded domain in Rn,a,b>0.First,we give some new definitions of lower and upper solutions for the problem,establish the method of lower and upper solutions and present some formulas of topological degree when the strict upper and strict lower solutions are defined.Next,we discuss the minus gradient flow of the functional corresponding to the Kirchhoff-type equations,present invariant sets of descent flow when the upper and lower solutions are listed and obtain the existence of the critical points for the functional.Finally,using the obtained theorems,we get the existence of at least three solutions for some kinds of Kirchhoff-type elliptic problems,one is positive,one is negative,and one sign-changing.In Chapter 2,using the method of sub-supersolutions and the fixed-point index method,we concerned with the sign-changing solutions and multiplicity results of the following problem where ?(?)RN(N?1),is a bounded and regular domain,and a,b?C?(?)(??(0,1))and a(x)>0,b(x)?0.First,we construct a new problem and prove the relative theorem of subsupersolutions.Then,we discuss the existence of positive and negative solutions for problem by applying the theorem.Based the fixed point index method,the existence of sign-changing solutions is established.