首页|Existence and asymptotics of normalized solutions for the logarithmic Schr?dinger system
Existence and asymptotics of normalized solutions for the logarithmic Schr?dinger system
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This paper is concerned with the following logarithmic Schrödinger system:{-Δu1+ω1u1=μ1u1 log u21+2p/p+q|u2|q|u1|p-2u1,-Δu2+ω2u2=μ2u2 logu22+2q/p+q|u1|p|u2|q-2u2,∫Ω|ui|2dx=ρi,i=1,2,(u1,u2)∈ H10(Ω;R2),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 logu2,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 Brézis-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 Schrödinger system with logarithmic perturbations.
logarithmic Schrödinger systemBrézis-Nirenberg problemnormalized solutionexistence and stabilitybehavior of solutions
Qian Zhang、Wenming Zou
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Department of Mathematical Sciences,Tsinghua University,Beijing 100084,China
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