Existences And Properties Of Solutions For Several Classes Of Nonlinear Partial Differential Equations Of Schr(?)dinger Type

The nonlinear partial differential equations of Schrodinger type is one of the most widely used equations in physics,appearing in many different branches.This doctoral dissertation studies the existences and properties of three kinds of nonlinear partial differential equations of Schrodinger type,including the exist,ence and non-existence of the ground state solution of the X-ray free electron lasers Schrodinger equation,and its asymptotic properties as the parameter for Coulomb like interaction tending to zero;the existences of solutions of the magnetic Schr(?)dinger equations with Hartree type nonlinearity and H1-critical power nonlinearity;the existence of the ground state of the fractional Schr(?)dinger-Poisson equation with "zero mass" and critical nonlinearity.This dissertation consists of five chapters.Chapter 1 is an introduction,where we give the background and current research status of the problems considered in this thesis,and show the main results and key ideas.Chapter 2 presents some lemmas and basic inequalities that we will use in this thesis.In Chapter 3,we consider the ground state of the following X-ray free electron laser Schrodinger equation(i▽-A(x))2u+V(x)u-μ/|x|u=(1/|x|*|u|2)u-K(x)|u|q-2u,x∈R3,which is a type of magnetic Schr(?)dinger equation,where A ∈ Lloc2(R3,R3)denotes the magnetic potential(the magnetic field B=curl A),μ∈R denotes the strength of the interaction between the electron beam and an atomic nucleus located at the origin,K ∈L∞(R3)is the parameter related to Fock approximation.In addition A and K are Z3-periodic and K is non-negative,q ∈(2,4).Based on a profile decomposition of the Cerami sequence in the magnetic Sobolev space HA1(R3)and a subtle analysis,we obtain the existence of the ground state solution for suitable μ≥0.When μ<0,we a.lso obtain the non-existence of the ground state solution.Furthermore,we describe the asymptotic behavior of the ground state sequence as μ→0-.In Chapter 4,we consider the existence of the following nonlinear magnetic Schrodinger equation with H1-critical nonlinearity(iε▽-A(x))2u+V(x)u=K(x)|u|4u+(1/|x|*|u|2)u,x∈R3,where V(x):R3→R is the external potential,K{x):R3→R is a parameter representing the strength of the three body interact,ion under Fock approximation and A(x)=(A1(x),A2(x),A3(x)):R3→R3 is the magnetic potential.In the 3-dimensional case,K(x)|u|4u is an H1-critical term.Under some suitable assumptions on V(x),K(x)and A(x),by proving the existence and compactness of(PS)sequence,we obtain the existence of solution when the parameter ε small enough.In Chapter 5,we consider the existence of the ground state of the following fractional Schrodinger-Poisson equation with critical exponent where s ∈(3/4,1),μ>0,p∈(3,s*)and s*=6/(3-2s)is the Hs critical exponent.Due to the absence of the quantity ‖u‖L22 in the energy functional,such a problem is called a "zero mass" problem.Because there is no information related to the L2-norm,the frame of the classical Sobolev space become invalid to deal with this equation,and we adopt the frame of Coulomb-Sobolev space.By the idea of the perturbation approach,we construct an asymptotic equation,and then based on the existence of solutions for the asymptotie equation,we finally obtain the ground state solution of the original equation.