The Existence Of Normalized Ground State Solutions For Schr(?)dinger Equations With Sobolev Critical Growth On The Heisenberg Group

As an important differential equation model,nonlinear Schrodinger equation plays an im-portant role in many mathematical physics problems such as nonlinear optics,and the existence of normalized ground state solution has been a research hotspot in recent years.The Heisen-berg group is the simplest sub-Riemannian manifold,which plays an important role in the fields of physics,geometry and astronomy.In this paper,the existence of normalized ground state solutions of the combined nonlinear Schrodinger equation with Sobolev critical growth on Heisenberg group is studied by means of variational method.The paper is divided into three parts:The first part is the introduction:This part describes the research progress of the nor-malized solution of the nonlinear Schr（?）dinger equation and the research background of the Heisenberg group,and gives the relevant definitions and conclusions on the Heisenberg group.In the second part,we apply the concentration-compactness principle and discuss the sub-additionality of the corresponding functional to prove the existence of the normalized ground state solution of the combined nonlinear Schrodinger equation with Sobolev critical growth and mass subcritical perturbation（2<q<2+4/Q）on the Heisenberg group H^N-△_Hu（ξ）=λu（ξ）+μ|u（ξ）|^q-2u（ξ）+|u（ξ）|^2_H*-2u,ξ∈H^N,where Q:=2N+2 is the homogeneous dimension of H^N,2_H^*:2Q/（Q-2）is the Sobolev critical index in H^N.In the third part,we use the method of subcritical approximation method to study the existence of normalized ground state solutions of the above combined nonlinear Schrodinger equation with Sobolev critical growth in terms of mass critical perturbation（q=2+4/Q）and mass supercritical perturbation（2+4/Q<q<2_H^*）.