The asymptotic behaviors of normalized ground states for nonlinear Schrödinger equations

• Published in: Nonlinear differential equations and applications Vol. 30; no. 3
• Main Authors: Zhang, Luyu, Zhang, Chengxiang
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.05.2023; Springer Nature B.V
• Subjects: Analysis; Asymptotic properties; Energy levels; Lagrange multiplier; Mathematics; Mathematics and Statistics; Rescaling; Schrodinger equation; Logarithmic nonlinearity; 35Q55; 35J20; Normalized ground states; Nonlinear Schrödinger equations; 35J60
• ISSN: 1021-9722; 1420-9004
• DOI: 10.1007/s00030-023-00853-z

In this paper, we study the relation between the least energy levels and between the minimizers of the following minimization problems E σ ( ρ ) = inf { 1 2 ∫ R N | ∇ w | 2 - 1 2 σ + 2 ∫ R N | w | 2 σ + 2 | ∫ R N w 2 = ρ } and Z ( ρ ) = inf { 1 2 ∫ R N | ∇ w | 2 - 1 2 ∫ R N w 2 log w 2 | ∫ R N w 2 = ρ } . We show that as σ → 0 + , the minimizers for E σ ( ρ ) , after rescaling, converge to the minimizers of Z ( ρ ) . Besides, we also give estimates for E σ ( ρ ) and the corresponding Lagrange multiplier when σ is small.