On The Existence And Multiplicity Of Solutions For Kirchhoff Type Equations In R~3

In this paper,we mainly study the existence and multiplicity of positive solution for Kirchhoff equations in R³.The thesis consists of four chapters:In Chapter One,we summarize the background of the related problems and state the main results of the present thesis.We also give some notations used in the whole thesis.In Chapter Two,we study the following Kirchhoff type problems:where the constants a,b>0,3<p<5,V?x?and Q?x?are two potentials in C_loc ??R³?? L_??R³?.By comparing the decay rate of V?x?and Q?x?,we first obtain two theorems stating the existence of positive ground states.On the other hand,the uniqueness of the positive radially symmetric solution of our problem is also analyzed when V?x?and Q?x?are constants.Under certain assumptions on potential Q?x?,we further prove the existence of positive bound states by using a linking argument with a barycenter map restricted on a Nehari manifold.In Chapter Three,we are concerned with the multiplicity of positive solutions for the Kirchhoff type problem:where e>0 is a parameter,a,b>0 are constants,p ??2,6?,and Q?x?? C?R³?is a nonnegative function.We show how the profile of Q?x?affects the number of positive solutions when ? is sufficiently small.Similar problem for Kirchhoff equation with Sobolev critical nonlinearity is considered in Chapter Four:where ?>0 is a parameter,a,n>0 are constants,p ??4,6?,and Q?x???R³?is a nonnegative function.In Chapter Five,we are concerned with the following Kirchhoff type equation:-?a+ b ? R³ |?u|2dx??u +?1 + ?h?x??u = |u|p-2u in R³,where a,b>0 are constants,?>0 is a parameter,2<p<6,and h?x?is a positive function in C2?R³?satisfying h?x?? 0 from above as |x| ??.Under certain assumptions on h?x?,we prove the existence of least energy solution when A>0 is sufficiently small.