The Existence Of Solutions To Kirchhoff-type Equations In The Whole Space

With the development of modern science and technology,the nonlinear functional anal-ysis has become an important part of mathematics.The nonlinear functional analysis plays a subtle role in many subjects,such as mathematics and physics.Therefore,it has attracted the attention of many mathematicians.As an important branch of nonlinear functional analysis,Kirchhoff equation has an im-portant influence in many different fields.Because the same Kirchhoff equation has many different results under different conditions,more and more scientists are interested in Kirch-hoff equation.Scientists get the solution of Kirchhoff type equation by using mountain pass theorem and surround theorem.At the same time,they also make an important contribution to Kirchhoff type equation.Because there are many different solutions of Kirchhoff type equation,we have strong curiosity about it.We study Kirchhoff type equation from subcritical problem to critical problem.In this paper,we prove the existence of the solution of the equation by using the mountain pass theorem,the Ekeland's variational principle and the Nehari manifold.This article is divided into four sections:Chapter 1 We briefly introduce the idea and theory of this paper.Chapter 2 We consider the following Kirchhoff-type equation-(?+?(?)|?u|2dx)?u+V(x)u=|u|p-2u+Q(x)f(u),in R3,where ?,?>0 are constants,4<p<6.Under appropriate assumptions on V,Q and ?,the existence of nontrivial solutions is proved by using the mountain pass theorem and the Ekeland's variational principle.Chapter 3 We study the following Kirchhoff type problem-(a+b?R4|?u|2dx)?u+V(x)u=K(x)u3+?f(u)in R4,where a,b>0 are constants,under suitable conditions on V,f and K.By using the generalized mountain-pass theorem,we obtain the existence and non-existence results.By using the Nehari manifold,we obtain a ground state solution for equation.Chapter 4 We study the following Kirchhoff-type equation where a?0.b>0.N?3,2*=2N/N-2,2<q<2*,?,?>0 are constants.By using variational method,under appropriate assumptions on a,b,?,?,we obtain the existence results.