Mass Concentration Of Blow-up Solutions For Nonlinear Schr(?)dinger Equations With Potentials

Quantum mechanics is the special theory to study the motion laws of microscopic particles,which has always been at the frontier field of physics since the 20th century.Schr(?)dinger equations are the basic kinetic equations of quantum phenomena,and vari-ous nonlinear Schr(?)dinger equations with and without potentials are the important con-tents of quantizing theory.Therefore,the mathematical research on nonlinear Schr(?)dinger equations has always been a hot subject in the field of mathematical physics.Nonlinear Schr(?)dinger equations are a class of typical wave equations with dispersion effect,which are general equations used in many branches of physics.When the disper-sion effect and the nonlinear effect reach balance,the standing waves are produced.The existence of standing waves is a typical characteristic of nonlinear Schr(?)dinger equations and plays an important role in the characterization of their solutions.From the research point of view of mathematical physics,whether the presence of standing wave solutions of stability in mathematics is an important criterion for judging the physical value of non-linear Schr(?)dinger equations,and nonlinear Schr(?)dinger equations with potentials more accurately reflect the nature of physics.Therefore,the study of the dynamic properties of nonlinear Schr(?)dinger equations with potentials is a significant content in quantum phe-nomena with the main feature of the standing wave solutions.This dissertation mainly focuses on the mass concentration of blow-up solutions for three types of nonlinear Schr(?)dinger equations with potentials,such as the nonlinear Schr(?)dinger equation with partial confinement,the nonlinear Schr(?)dinger equation with inverse-square potential and the nonlinear Schr(?)dinger equation with double potentials.The main contents of this dissertation are as follows:1.We study the mass concentration of blow-up solutions for the nonlinear Schr(?)dinger equation with partial confinement in R~2.Firstly,by using the variational characteristic of the classical nonlinear scalar field equation and the Hamilton conservations,we get the sharp threshold for global existence and blow-up of the nonlinear Schr(?)dinger equation with partial confinement on mass in two-dimensional space.Then,in terms of the re-lated compactness result and the variational characteristic of the ground state of nonlinear scalar field equation,we prove the mass concentration of blow-up solutions for the non-linear Schr(?)dinger equation with partial confinement.Finally,we discuss weak L~2(R~2)limit and the limiting profile of blow-up solutions with small super-critical mass for the nonlinear Schr(?)dinger equation with partial confinement.2.We investigate the mass concentration of blow-up solutions for the nonlinear Schr(?)dinger equation with inverse-square potential.According to the existence of mini-mal mass blow-up solutions for the nonlinear Schr(?)dinger equation with inverse-square potential and the refined compactness result,we discuss the mass concentration of blow-up solutions for the nonlinear Schr(?)dinger equation with inverse-square potential in detail.3.We consider the mass concentration of blow-up solutions for the nonlinear Schr(?)dinger equation with double potentials.Firstly,by solving a variational problem concerning the equation,we get a Gagliardo-Nirenberg inequality with the best constant.Then,according to the Gagliardo-Nirenberg inequality,we prove the global existence of solutions and the instability of standing waves for the nonlinear Schr(?)dinger equation with double poten-tials.Finally,we discuss the mass concentration of blow-up solutions for the nonlinear Schr(?)dinger equation with double potentials by using a refined compactness result,which can be proved by the profile decomposition of bounded sequences in H~1(R^N).4.We study the existence and the uniqueness for the L~2(R^N)-critical nonlinear Schr(?)dinger equation with double potentials.Based on the variational characteristic of the ground state of the corresponding nonlinear scalar field equation and the concentration compactness principle,we first obtain a new compactness result.Then by combining with the pseudo-conformal transformation of the nonlinear Schr(?)dinger equation with double potentials and its related symmetry properties,we construct the blow-up solutions at the minimal mass threshold and further prove the uniqueness result on the minimal mass blow-up so-lutions.