Existence Of Solutions For Two Classes Of Nonlinear Elliptic Equations

In this thesis,we study the existence of weak solutions for critical nonlinear Choquard equa-tions with low-order term and the fractional Schrodinger equation with magnetic field.This thesis is divided into three chapters.In chapter 1,we summarize the background and recent developments of the related problems.Moreover,the main results of the thesis.In chapter 2,we study the following critical nonlinear Choquard equations with low-order term:-?u+a(x)u?(|x|-?*|u|2?*)|u|2?*-2u+q(x)|u|p-1u,(0.0.3)where 1<p<2*-1,N?4,with 2*=2N/N-2,2?*=2N-?/N-2,a(x)and q(x)are two posithive continuous functions.By using the variational method,we study the existence of solution for the limit equation at first,and then combing the global compactness result and the mountain pass theorem,we obtain the existence of the solution for the given equation.In Chapter 3,we consider the following fractional Shrodinger equation with magnetic field:(-?)sAu+V?(x)u=f(x)|u|q-2+g(x)|u|p-2u,x ? RN,(0.0.4)where 0<s<1,N>2s,1<q<2<p<2s*with 2s*=2N/N-2s,the potential satisfies V?(x)=?V+(x)-V-(x),?>0 is a parameter,magnetic field A?C0,?(RN,RN)with(??(0,1]),and u:RN?C.Here(-?)sAu is the fractional magnetic Laplacian operator.By considering the minimizers on Nehari manifold,we obtain the existence of solutions for function(0.0.4).