Studies On The Properties Of The Solutions Of L~2 Critical Non-Local Schr?dinger Equations

This doctoral thesis studies the nonlocal Schrodinger equations of the following form where A is a differential operator,G(u)is the nonlinear term,(t,x)∈ R×RN,and N≥1.We are concerned with two kinds of non-local Schrodinger equations and our main results can be summarized as the following two parts1.(?),G(u)=-(|u|2/N u+k|u|pu).In this situation,the equation is the half-wave equation i(?)tu + Du + u2/Nu + kuρu = 0,which is a special case of the fractional Schrodinger equation,and is also the degen-erated case for the semi-relativistic Schrodinger equation with zero mass.Usually,i(?)t+D is called a half-wave operator.For the case κ=0:When N=2,we obtain the existence of finite-time blowup solution with ground state mass,which satisfies ‖u0‖2=‖Q‖2(ground state mass conservation)and E(u)=E(u0)(energy conservation),where Q is the unique positive radial ground state solution of the equation DQ+Q=|Q|2/NQ.The blowup speed is given by ‖D1/2u(t)‖L2～C(u0)/|t| as t→0-.When N=3,we also obtain the similar existence of finite-time blowup solutions with ground state mass and the blowup speed.For the case κ=1:Assume 0<p<2/N and N≥2.If the initial data ‖u0‖2<‖Qv‖2,where Qv is a solution of the equation (?)-Δuv+i(v(?))uv-uv2/Nuv=-uv,we obtain the existence of traveling wave solutions of the form u(t,x)=eitμΨv(x-vt)with 0<|v|<1.In addition,when N=2,3,4,this traveling solitary waves are orbitally stable.2.A=-Δ,and G{u)=V(x)u-a(1/|x|γ*|u|2)u is a non-local nonlinear term.In this situation,the corresponding equation is called a Schrodinger equation with Hartree-type nonlinear term or Hartree equation7 where a(1/|x|γ*|u|2)u in called a Hartree term,which describes some long-range interactions between particles in non-relativistic quantum mechanics.When N≥3 and a>a*=‖Q‖L22,where Q is the positive radial ground state solution of equation-Δu+u-(1/|x|*|u|2)u=0,by virtue of the constrained variational method and energy estimates,we give a detailed description on the concentration and symmetry breaking phenomena for the standing wave solutions of the form u(x)eiλt as γ↗2(where 2 is the L2 critical exponent).