Normalized Solution for the Logarithmic Schrödinger System Normalized Solution for the Logarithmic Schrödinger System

• Published in: The Journal of geometric analysis Vol. 35; no. 7; p. 215
• Main Authors: Liu, Tianhao, Peng, Xueqin, Zou, Wenming
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.07.2025; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Convex and Discrete Geometry; Defocusing; Differential Geometry; Dynamical Systems and Ergodic Theory; Fourier Analysis; Global Analysis and Analysis on Manifolds; Lagrange multiplier; Logarithms; Mathematics; Mathematics and Statistics; Scalars; 35J10; 35J50; Stability/instability and well-posedness; Asymptotic behavior; 35A01; Normalized solution; Logarithmic Schrödinger system; Existence and multiplicity
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-025-02053-w

This paper is concerned with the existence and properties of normalized solutions to the following logarithmic Schrödinger system - Δ u + λ 1 u = μ 1 u log u 2 + θ α | u | α - 2 | v | β u in R N , - Δ v + λ 2 v = μ 2 v log v 2 + θ β | v | β - 2 | u | α v in R N , ∫ R N u 2 d x = a 2 and ∫ R N v 2 d x = b 2 , where N ≥ 2 , a , b > 0 , μ 1 , μ 2 , θ ∈ R \ 0 , and the exponents α > 1 , β > 1 satisfy the Sobolev critical and subcritical conditions 2 < α + β < ∞ , when N = 2 , 2 < α + β ≤ 2 ∗ , when N ≥ 3 . The parameters λ 1 , λ 2 ∈ R will arise as Lagrange multipliers that are not prior given. For the focusing case μ 1 , μ 2 > 0 , we establish various results concerning the existence, multiplicity, and stability/instability of normalized solutions when θ > 0 . Furthermore, we demonstrate that as θ → 0 + , these normalized solutions of the logarithmic Schrödinger system converge to constant multiples of a specific solution with physical significance, known as the Gausson , which is the unique positive ground state solution of the logarithmic scalar field equation - Δ u = u log u 2 in R N , u ∈ H 1 ( R N ) . Besides, we also give a criteria for global existence and finite time blow-up in the associated dispersive system. Finally, for the defocusing case μ 1 , μ 2 < 0 , we prove a nonexistence result when θ < 0 and find a radially symmetric sign-changing normalized solution when θ > 0 . This paper combines several approaches and gives a rather complete picture of the logarithmic Schrödinger system.