Research On The Existence Of Solutions Of The Weakly Coupled Elliptic System

In this thesis,we mainly study the existence and properties of solutions for the nonlinear weakly coupled nonlocal elliptic equations.On the one hand,we use the variational method to study the existence of solutions as well as the relevant properties for functional critical point of the equation.On the other hand,we use the Morse theory to get a ground state solution for equations and its Morse index is 1.This thesis includes the following four Chapters.In Chapter One,we summarize the background of the related problems and state the main results of the present thesis.In Chapter Two,we introduce notations and preliminary results.In Chapter Three,we study the existence and properties of the solutions of the following nonlinear elliptic systemswhere ??R3,?i,?i>0(i= 1,2).According to solutions of the single elliptic equation,we compare the corresponding functional energy by variational method.Then,we obtain ground and bound state solutions of the system when ? satisfied some conditions.Finally,we get the existence condition of the positive ground state solution for general systems.Compared with results in[51],our conditions are better than[51].In the Chapter Four,we consider general weakly coupled systems the existence and nonexistence of solutions of the equations can be obtained.For one thing,we prove the existence a positive ground state solution using Morse index.For another thing,we find the exact conditions to guarantee the existence of positive ground state solution of systems with k = 2...