Existence Of Solutions To Two Classes Of Kirchhoff Type Equations

The Kirchhoff type equation is one of the typical equations in Elliptic partial differential equations,which was put forward by Kirchhoff when he studied the length changes of the string.In recent years,the Kirchhoff type equation is widely used in physics,aerospace tech-nology,biotechnology and other fields.Because kirchhoff equation can get different results under different conditions,there are a lot of literatures on the research methods and results of this equation.For example,the solution to Kirchhoff equation is obtained by using the invariant sets of descent flow,mountain pass theorem,morse theory and so on.In this pa-per,we obtained the existence of the solutions to Kirchhoff equation via fountain theorem,quantitative deformation lemma and G-linking theorem.This paper is divided into three chapters:Chapter 1 We introduced the relevant background knowledge and the theorem in this paper.Chapter 2 We studied the kirchhoff equation with parameters in the whole space,as shown above:where a is a positive constant,? is a parameter,V and f satisfied the appropriate conditions.Using the fountain theorem and quantitative deformation lemma to prove the existence of infinitely many solutions and sign-changing solutions to the above equations.Chapter 3 We studied the kirchhoff equation with an asymptotically 4-linear nonlin-earity,as follows:where a,b>0,f is an asymptotically 4-linear as |u|??,we obtained the weak solutions via G-linking theorem and mountain pass theorem.