Research On A Class Of Nonlinear Schr?dinger Equations

In this paper,we mainly study the existence of solutions for a quasilinear Schr?dinger equations and nonlinear Schr?dinger equations with sign-changing electromagnetic fields,The thesis consists of three chapters:In chapter One,we summarize the background of the related problems and state the main results of the present thesis.We also give some preliminary results and notations used in the thesis.In chapter Two,we consider the existence of positive solutions for the follow-ing quasilinear Schr?dinger equation where ? is a bounded smooth domain in RN,paramater k>0,p ?(2,22*)\{4},and h(x,s,?)satisfy some given conditions.By using the priori estimates and fixed point theorems,we proved the existence of positive solutions of the problem(1).Our results not only provided a method that can be used to prove the existence of positive solution for the quasilinear Schr?dinger equations without variational structure,but also gave a priori bounds of positive solutions.In chapter Three,we study the existence of complex-valued solutions for the following nonlinear Schr?dinger equations with sign-changing electromagnetic fields(?/i-A(x))2u+V(x)u=f(x,|u|2)u,u:RN?C,(2)where i is an imaginary unit,A(x)=(A1(x),A2(x),…AN(x))represents the electromagnetic fields,Aj(x)?C1(RN,R)(j=1,2,…N).The electric potential V(x)? C(RN,R)may change sign.Considering A(x),V(x)and f(x,|u|2)satisfy-ing certain conditions,The existence of complex-valued solutions for the problem(2)is obtained by making use of the linking theorem under(C)c conditions.Our results extend the Ding and Szulkin's in[15]where they study the problem(2)in the case of A(x)=0.