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 + ku?u = 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 ?=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 ?=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 (?)-?uv+i(v(?))uv-uv2/Nuv=-uv,we obtain the existence of traveling wave solutions of the form u(t,x)=eit??v(x-vt)with 0<|v|<1.In addition,when N=2,3,4,this traveling solitary waves are orbitally stable.2.A=-?,and G{u)=V(x)u-a(1/|x|?*|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|?*|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-?u+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)ei?t as ??2(where 2 is the L2 critical exponent).