Existence And Multiplicity Of Normalized Solutions For Nonlinear Schr(?)dinger Equations

This dissertation focuses on the study of the nonlinear Schr (?)dinger equation with the mass constraint:where N ? 1,f ? C(R,R),m > 0 is a given constant and ? ? R arises as a Lagrange multiplier.In Chapter 1,we give a brief account of the background and the current states of the problem,and state the main results of this dissertation.In Chapter 2,we are concerned with the mass subcritical case.Under general mass subcritical assumptions on f,we show the existence of one nonradial solution for any N ? 4,and obtain multiple(sometimes infinitely many)nonradial solutions when N = 4or N ? 6.In particular,all these solutions are sign-changing.Besides,we also establish a multiplicity result of radial solutions for any N ? 2.In Chapter 3,we consider the mass supercritical case.Assuming on f weakened mass supercritical conditions,including a monotonicity condition of Pohozaev type,we show the existence of ground states and reveal the basic behavior of the ground state energy as m > 0 varies.In particular,to overcome the compactness issue when looking for ground states,we develop robust arguments which we believe will allow treating other mass constrained problems in general mass supercritical settings.Under the same assumptions,we also obtain infinitely many radial solutions for any N ? 2 and establish the existence and multiplicity of nonradial sign-changing solutions when N ? 4.In the last chapter,we summarize the main works of this dissertation and propose several open problems for further study.