Existence And Properties Of Solutions For Some Kirchhoff Equations

In this paper,the existence,uniqueness and concentration of solutions of several kinds of Kirchhoff equations are discussed by using variational method and energy estimate techniques.The main contents are as follows:In Chapter 1,we mainly introduce some background and research status on Kirchhoff equations.In Chapter 2,we give some basic knowledge related to the problems discussed in this paper.In Chapter 3,we discuss the existence and uniqueness of ground state solutions of the following p-Kirchhoff equation:(?)(P)where a,b>0,1<p<n,and p<s<p*:=np/(n-p).The corresponding energy functional of equation(P)is#12 By applying the variational method together with some energy estimates,we prove the existence and uniqueness of global minima or critical points of mountain path type for the energy functional Ep(·)on the constrained manifold#12In addition,we show that these critical points satisfy the equation(P)and are actually optimizers of some Gagliardo-Nirenberg inequality.In chapter 4,we study the following Kirchhoff equation with a singular parameter(?)a,b>0,N≥3,(K)where g(x,u)=K(x)|u|2*-2 u or g(x,u)=-V(x)u+|u|p-2 u(2<p<2*)is critical or subcritical growth.By introducing suitable scalings,(K)is transformed into a semilinear elliptic equation coupled with an algebraic equation of the scaling parameter.Upon which,we prove the existence of multi-peak solutions of(K)which concentrating around the critical points of K(x)or V(x).Moreover,we prove that(K)has only trivial solution if N≥4 and b>0 is suitably large.In chapter 5,we summarize the main conclusions and innovations of this paper,and list some problems that we intend to study in the future.