Studies On The Existence Of Solutions For Some Quasilinear Schr?dinger Equations And A Class Of Schr?dinger-Poisson Systems

In this thesis,we study the existence of solutions for two classes of quasilinear Schrodinger equations and a class of Schrodinger-Poisson systems.This dissertation con-sists of four parts.In Chapter 1,we summarize the background of the related problems and state the main results of present thesis.We also introduce the main results and disposal methods of the dissertation.Moreover,we give some preliminaries involved in this dissertation.In Chapter 2,We study a class of quasilinear Schrodinger equations in one-dimensional space:-u"+V(x)u-(|u|2)"u=f(u),x?R.where the potential is indefinite of sign so that the Schrodinger operator possesses a finite dimensional negative space,without requiring the working space can be compactly embed-ded into LP(R).By using the Morse theory,we obtaexponent or with a nonsymmetric term[J].Trans.Amer.Math.Soc.,1989,323:877-in at least one nontrivial solution for the problem.In Chapter 3,We study a class of quasilinear Schrodinger equations with indefinite potentials:-?u+V(x)u-u?(|u|2)=g(u),x?RN.where g(u)is 4-superlinear.The potential is indefinite of sign so that the Schrodinger op-erator possesses a finite dimensional negative space,hence the variational functional does not satisfy the mpuntain pass geometry.By a local linking argument and Morse theory,we obtain a nontrivial solution for the problem.In case that g is odd,we get an unbounded sequence of solutions.In Chapter 4,We study a class of nonlinear Schrodinger-Poisson systems:Under suitable conditions on V,K,g and h,when 1<s<6,using the method of construct-ing truncated function,we obtain two nontrivial solutions for the problem.When g(x,·)is odd and 6<s<? we obtain infinitely many solutions for the problem.