Two Types Of Nontrivial Solutions For A Class Of Elliptic Equations

More recently, many classes of nonlinear elliptic equations emergence with the development of nature science and engineering,so the study of the existence and multiple solutions to the nonlinear elliptic equations becomes an important subject. In this paper,we primarily study the existence of solutions to the two classes of nonlinear elliptic equations by applying the invariant sets of descending flow and moutain pass of variational methods,and obtain their positive solutions,negative solutions,sign-changing solutions and multiple nontrival solutions in some certain conditions,and these new conclusions have developed the previous results.The full text divides into three chapters:In chapter one,we begin with the research background,the development process and some concise review of the basic definitions and theorems.In chapter two,we discuss the existence of solutions to the semilinear elliptic equation where Ω is a bounded domain inRN with smooth boundary (?)Ω.This class of nonlinear elliptic equation is a very important type of nonlinear partial differential equation,so more and more mathematicians, biologists and physicists have generated a strong interest in the study, and obtained many useful results. In this chapter,we obtain sign-changing solutions,positive solutions and negative solutions of the equation where the variational functional satisfies the (PS) compactness condition and without the (PS) compactness condition respectively.In chapter three,we continue to utilize the invariant sets of descending flow methods to discuss the existence of solutions to the Kirchhoff equation where Ω is a bounded domain in RN with smooth boundary (?)Ω,and α, b>0.This chapter has been studied under weaker conditions than the existing literature on characteristic of the variational functional,and multiple solutions of this equation are obtained by the related local linking theory.