Existence and asymptotics of normalized solutions for the logarithmic Schr?dinger system

This paper is concerned with the following logarithmic Schr?dinger system:■where Ω=RN or Ω?RN（N≥3） is a bounded smooth domain,and ωi∈R,μi,ρi> 0 for i=1,2.Moreover,p,q≥1,and 2≤p+q≤2*,where 2*:=2N/N-2. By using a Gagliardo-Nirenberg inequality and a careful estimation of u log u2,firstly,we provide a unified proof of the existence of the normalized ground state solution for all 2≤p+q ≤2*.Secondly,we consider the stability of normalized ground state solutions.Finally,we analyze the behavior of solutions for the Sobolev-subcritical case and pass to the limit as the exponent p+q approaches 2*.Notably,the uncertainty of the sign of u log u2 in（0,+∞） is one of the difficulties of this paper,and also one of the motivations we are interested in.In particular,we can establish the existence of positive normalized ground state solutions for the Brézis-Nirenberg type problem with logarithmic perturbations（i.e.,p+q=2*）.In addition,our study includes proving the existence of solutions to the logarithmic type Bréis-Nirenberg problem with and without the L2-mass.constraint ∫Ω|ui|2dx=ρi（i=1,2） by two different methods,respectively.Our results seem to be the first result of the normalized solution of the coupled nonlinear Schr?dinger system with logarithmic perturbations.