On Nonlocal Choquard System with Hardy–Littlewood–Sobolev Critical Exponents.

Standing wave solutions of the following Hartree system with nonlocal interaction and critical exponent are considered: - (a + b ∫ Ω | ∇ u | 2) Δ u = h (x) ∫ Ω | v (y) | 2 μ ∗ | x - y | μ d y | u | 2 μ ∗ - 2 u + f λ (x) | u | q - 2 u , i n Ω , - (a + b ∫ Ω | ∇ v | 2) Δ v = h (x) ∫ Ω | u (y) | 2 μ ∗ | x - y | μ d y | v | 2 μ ∗ - 2 v + g σ (x) | v | q - 2 v , i n Ω , u , v ≥ 0 , i n Ω , u , v = 0 , o n ∂ Ω , <graphic href="12220_2022_959_Article_Equ67.gif"></graphic> where 1 < q < 2 , 2 μ ∗ = 2 N - μ N - 2 is the upper critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. We study the effect of nonlocal interaction on the number of solutions in the case of general response function Ψ (x) = | x | - μ (0 < μ < N) , which possesses more information on the mutual interaction between the particles. When parameters pair (λ , σ) belongs to a certain subset of R 2 , we prove the existence, nonexistence and the limit behavior of the nonnegative vector solutions depending on parameters. In the special case of q = 2 , existence of nonnegative solution is also established. Our work extends and develops some recent results in the literature. [ABSTRACT FROM AUTHOR]