Research On The Existence Of Normalized Solutions In Nonlocal Elliptic Equation And Nonhomogeneous NLS Equations

This thesis mainly studies the existence of the normalized solutions about nonlocal elliptic equation and nonhomogeneous nonlinear schr(?)dinger equations on bounded domian.The normal-ized solution in here refers to the solution of the equation(s)under the L~2mass constraint.The nonlocal elliptic equation was originally derived from the discovery of D.R.Hartree about the Hartree equation,which treated each electron as moving in the average potential field provided by the rest of the electrons,and gave the equation of motion for each electron.The source of the nonhomogeneous nonlinear schr(?)dinger equations were the E.P.Gross's discovery about the Gross-Pitaevskii-type NLS equation,it is great significance in the fields of plasma physics,fluid dynamics,nonlinear optics,and Bose-Einstein condensation and so on.The first part mainly studies the existence of the solution about the nonlocal elliptic equation with two mass constraints on bounded domain.According to the idea of variational method,this problem is usually transformed into finding the minimax value problem of the energy functional.Since the partial term of the energy functional corresponding to the elliptic equation is limited by the mass constraints,it is only necessary to consider whether the nonlocal term minimax value problem is achieved.In this thesis,the modified Gagliardo-Nirenberg inequality is used to process the nonlocal term,and finally the normalized solution of the nonlocal elliptic equation is obtained.In addition,according to the method proposed by Ambrosetti and Prodi,the space and map are constructed,and then the asymptotic expansion of the normalized solution is obtained by using the singularity theorem.The second part mainly studies the existence of the normalized solutions of nonhomogeneous nonlinear schr(?)dinger equations on bounded domain.In the case of L~2-subcritical and L~2-critical,because the energy functional is coercive,the global solution of the nonhomogeneous NLS equa-tions is found by using the minimax principle in variational method;in the case of L~2-supercritical and Sobolev-subcritical,find out the local solution of the nonhomogeneous nonlinear schr(?)dinger equations by providing some appropriate conditions about the parameters;in the Sobolev-critical case,the functional is no longer weakly lower semi-continuous,and the local solution of the ho-mogeneous NLS equations is found by restoring the compactness about the minimizing sequence.