Prescribed Mass Solutions to Schrödinger Systems With linear Coupled Terms

• Published in: The Journal of geometric analysis 2023-11, Vol.33 (11), Article 347
• Main Authors: Chen, Haixia, Yang, Xiaolong
• Format: Article
• Language: English
• Subjects: Abstract Harmonic Analysis; Asymptotic properties; Convex and Discrete Geometry; Differential Geometry; Dynamical Systems and Ergodic Theory; Fourier Analysis; Geometry; Global Analysis and Analysis on Manifolds; Lagrange multiplier; Mathematics; Mathematics and Statistics
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-023-01405-8

We consider the existence of normalized solutions to the following nonlinear Schrödinger system - Δ u + λ 1 u = μ 1 | u | p - 2 u + β v in R N , - Δ v + λ 2 v = μ 2 | v | q - 2 v + β u in R N , under the mass constraints ∫ R N | u | 2 d x = a 1 2 and ∫ R N | v | 2 d x = a 2 2 , where 2 < p ≤ 2 + 4 N ≤ q ≤ 2 ∗ , β , μ 1 , μ 2 > 0 , a 1 , a 2 > 0 , and λ 1 , λ 2 ∈ R appear as Lagrange multipliers. Under different character on p ,  q with respect to the mass critical exponent, we prove several existence results and precise asymptotic behavior of these solutions as ( a 1 , a 2 ) → ( 0 , 0 ) . These cases present substantial differences with respect to purely mass subcritical or mass supercritical situations.