Normalized Solutions to Fractional Mass Supercritical Choquard Systems

• Published in: The Journal of geometric analysis 2024-04, Vol.34 (4), Article 104
• Main Authors: Guo, Zhenyu, Jin, Wenyan
• Format: Article
• Language: English
• Subjects: Abstract Harmonic Analysis; Convex and Discrete Geometry; Differential Geometry; Dynamical Systems and Ergodic Theory; Fourier Analysis; Global Analysis and Analysis on Manifolds; Lagrange multiplier; Mathematics; Mathematics and Statistics
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-024-01548-2

In this paper, we mainly study the fractional Choquard systems with a local perturbation under the mass constraints: ( - Δ ) s u = λ 1 u + ( I α ∗ | u | 2 α , s ∗ ) | u | 2 α , s ∗ - 2 u + μ 1 | u | p - 2 u + β r 1 | u | r 1 - 2 u | v | r 2 in R N , ( - Δ ) s v = λ 2 v + ( I α ∗ | v | 2 α , s ∗ ) | v | 2 α , s ∗ - 2 v + μ 2 | v | q - 2 v + β r 2 | v | r 2 - 2 v | u | r 1 in R N , ‖ u ‖ L 2 ( R N ) 2 = a 1 2 and ‖ v ‖ L 2 ( R N ) 2 = a 2 2 , where ( - Δ ) s u is the fractional Laplacian, I α ( x ) is the Riesz potential, 0 < α < min { N , 4 s } , and N > 2 s , s ∈ ( 0 , 1 ) , λ 1 , λ 2 ∈ R N are unknown constants, which will appear as Lagrange multipliers, μ 1 , μ 2 , β , a 1 , a 2 > 0 , r 1 , r 2 > 1 , p , q and r 1 + r 2 ∈ ( 2 + 4 s N , 2 s ∗ ] . 2 s ∗ = 2 N N - 2 s is the fractional critical Sobolev exponent and 2 α , s ∗ = 2 N - α N - 2 s is the fractional Hardy–Littlewood–Sobolev critical exponent. Firstly, if p , q and r 1 + r 2 < 2 s ∗ , we obtain the existence of positive normalized solution when β is big enough. Then, for the case of p = q = r 1 + r 2 = 2 s ∗ , we may obtain the nonexistence of positive normalized solution.