Existence And Concentricity Of Solutions Of Coupled Nonlinear Schr(?)dinger Equations

In this thesis,we use methods such as constrained variation and Nehari manifolds to study the existence and concentration of completely non-trivial solutions of coupled nonlinear Schr(?)dinger equations under the attractive condition.The full text is divided into five chapters,and the specific content is arranged as follows:In Chapter 1,the theoretical background of the coupled nonlinear Schr(?)dinger equations studied in this thesis and the research status of the considered problems are introduced.In Chapter 2,we study the existence of positive ground state solutions of the coupled Choquard equation with weighted potential and lower critical exponent using the constrained variational method.First,the existence of the ground state solution of the problem is transformed into a constrained variational problem,the tightness is usually missing due to the presence of the lower critical exponent,and the maximization sequence of this constrained variational problem may not converge strongly.To verify the strong convergence of the maximization sequence,the Brezis-Lieb Lemma is used to make a decomposition of the maximization sequence of this problem.Next,we combine it with the maximum energy estimate of the constrained variational problem,we can prove that the maximization sequence has a subset of strong convergence.Finally,by using the Lagrange multiplier principle and excluding the existence of semi-trivial solutions,a positive ground state solution is obtained.Moreover,by using Hardy’s inequality and one Pohozˇaev identity,a non-existence result of non-trivial solutions is also considered.In Chapter 3,the existence of completely nontrivial solutions of the nonlinear Schr(?)dinger equation system with constant coefficients problem is investigated by using the Nehari manifold method.On the one hand,it is proved that the manifold is canonically bounded,that is,the energy generalization is restricted to the critical point where the constraint is minimal and unbounded;on the other hand,the existence of completely nontrivial solutions to the system of equations is obtained using the Schwartz symmetry method.In Chapter 4,we study the existence and concentration of completely nontrivial solutions to the nonlinear Schr(?)dinger equation system with singular perturbation problem,under the attraction condition.First,the main idea is to find the solutions of the equations in the vicinity of the solution set of the limit problem,the goal is to make a translational transformation of the solutions of the limit problem with a small perturbation;then,we find the critical point of the energy generalization on the manifold,and the critical point corresponds to the local minima of the energy of the limit problem;immediately,we estimate the energy of the energy generalization on the manifold,and then the existence of the PS sequence is obtained by proving that the minima can only be taken internally;finally,we use the Ekeland variational principle to obtain the minima.In Chapter 5,we summarize the results of this thesis and make some discussions.