Existence Of Positive Solutions To A Class Of Schr(?)dinger Equations With Two Nonlocal Terms And The Critical Exponent

Nonlocal elliptic equations are widely used in the fields of geometry,physics,biology and so on,and are a kind of widely concerned problems in the field of nonlinear analysis in recent years.Such problems are characterized by the fact that the equations are no longer valid in the pointwise sense.In this thesis,we are mainly concerned with the existence of solutions to the following Schrodinger equations with two nonlocal terms via variational method-?u+(u2*1/|4?x|)u=?(1/|x|?*|u|p+1)|u|p-1u+|u|4u,in R3 where ?>0 and ?>0.There are some difficulties in applying variational methods to the above equation.The first one is that the nonlinear term has Sobolev critical exponent,which affects the compactness of the energy functional.The second difficulty is that the range of parameters A and p will affect the properties of the energy functional and thus the existence of the solution.In addition,the influence of the two nonlocal terms on the energy functional geometry needs to be analyzed carefully.Firstly,we introduce the background of the equation.Secondly,in the case of p ?(2,5),we establish the compactness of the energy functional via the perturbation method under the condition that ??(0,2)and 6(p+1)/6-?[3,6],and then we obtain a positive ground state solution of the above equation.Finally,for p?(11/7,2],??(0,2),r=6/6-? satisfies r(p+1)2-6/3r?(0,1),combining the truncation method with the concentration compactness principle,we show the existence of a positive radial symmetry to the above problem.