Study On The Existence Of Solutions For Some Classes Of Kirchhoff Type Equations

Kirchhoff type equation is a nonlinear elliptic equation with nonlocal terms,which is one of the basic equations in physics.Many qualitative properties of the equation can explain the physical significance of many nonlinear models in physical science,and have very important applications in physics,such as the effect of electromagnetic field on the motion of electric current.In the present thesis,we consider the existence of solutions for some classes of Kirchhoff type equations by using variational method and critical point theory.The thesis consists of six chapters and it is organized as follows.In Chapter 1,we introduce the research background,research status and main works of Kirchhoff type equations,and give some basic knowledge of this paper.In Chapter 2,when assumptions on the nonlinear function just in a neighborhood of=0,we prove the existence of nontrivial solutions for a class of nonlinear Kirchhoff-type equations by using truncation function,Mountain Pass Theorem and L~?(R~3)estimate.In Chapter 3,when the nonlinear function satisfies different conditions,we prove the existence of nontrivial solutions and sign-changing solutions for two classes of nonlinear Kirchhoff-type equations with critical or supercritical growth via truncation function and Mountain Pass Theorem.In Chapter 4,with the nonlinear function is a continuous function and just superlinear in a neighborhood of=0.We prove the existence of nontrivial solutions for a class of nonlinear fractional Kirchhoff type equation by using an appropriate truncation argument and the Mountain Pass Theorem.In Chapter 5,we prove the existence of sign-changing solutions for a class of nonlinear fractional Kirchhoff-type equations with critical or supercritical growth by combining an appropriate truncation argument with Moser iteration method,In Chapter 6,we summarize the main research results of this paper and propose the direction of future research.