Abstract
This paper is concerned with the following logarithmic Schrödinger 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 \({2^ * }: = {{2N} \over {N - 2}}\). 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 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.
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The authors are very grateful to the referees for their careful reading and for their very helpful comments and suggestions.
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Zhang, Q., Zou, W. Existence and asymptotics of normalized solutions for the logarithmic Schrödinger system. Sci. China Math. (2024). https://doi.org/10.1007/s11425-022-2172-x
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DOI: https://doi.org/10.1007/s11425-022-2172-x
Keywords
- logarithmic Schrödinger system
- Brézis-Nirenberg problem
- normalized solution
- existence and stability
- behavior of solutions