Existence Of Standing Wave Solutions For Two Kinds Of Schr(?)dinger Equations

In this thesis,we mainly study the existence of solutions for two types of Schrodinger equations,including positive bound state solutions of Schrodinger equation with saturable nonlinearity and normalized solutions of quasilinear Schrodinger equation,in which we consider two cases of constant zero function and harmonic function on the potential V(x)for the quasilinear Schrodinger equation,respectively.This paper is divided into four chapters as follows:In Chapter 1,we firstly introduce the research background and significance of the above two types of Schrodinger equation.After introducing the research status and main contents of the Schrodinger equation at home and abroad,we give some definitions,lemmas and other related preliminary knowledge that will be used in this paper.At the end of this chapter,we introduce the overall structure of this paper.In Chapter 2,we consider the following Schrodinger equation with saturable nonlinearity (?) where,N≥1,i is an imaginary number unit,and μ>0 is a coupling constant.We need to study when the intensity function V(x)satisfies certain conditions,the ground state solution of the equation does not exist,but a positive bound state solution of the equation is obtained through Nehari manifold method,the linking theorem and the topological degree theoryIn Chapter 3,we consider the quasilinear Schrodinger equation:(?) We discuss the potential function V(x)satisfies the following two cases:(?) By using the compact embedding in the radial symmetric function space to estimate the quasilinear part,combining the properties of convex function and Brizes-Lieb lemma,we obtain the normal solution of the quasilinear Schrodinger equation when the potential function V(x)satisfies(1)and the Lagrange multiplier is limited to a certain range.For(2),the normalze solution of the quasilinear Schrodinger equation can be easily obtained by using the method of energy estimation and compactness.In Chapter 4,We have summarized the results and raised questions worth exploring in response to the results of this article.