Positive ground states for coupled nonlinear Choquard equations involving Hardy–Littlewood–Sobolev critical exponent. (August 2019) - You, Song; Zhao, Peihao; Wang, Qingxuan
Abstract: In this paper, we consider the following coupled nonlinear Schrödinger equations with Choquard type nonlinearities: − Δ u + λ 1 u = μ 1 ( 1 | x | μ ∗ u 2 μ ∗ ) u 2 μ ∗ − 1 + β ( 1 | x | μ ∗ v 2 μ ∗ ) u 2 μ ∗ − 1, x ∈ Ω, − Δ v + λ 2 v = μ 2 ( 1 | x | μ ∗ v 2 μ ∗ ) v 2 μ ∗ − 1 + β ( 1 | x | μ ∗ u 2 μ ∗ ) v 2 μ ∗ − 1, x ∈ Ω, u, v ≥ 0 i n Ω, u = v = 0 o n ∂ Ω, where Ω ⊂ R N is a smooth bounded domain, 2 μ ∗ ≔ 2 N − μ N − 2 is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality, − λ 1 ( Ω ) < λ 1, λ 2 < 0, λ 1 ( Ω ) is the first eigenvalue of ( − Δ, H 0 1 ( Ω ) ), μ 1, μ 2 > 0 and β ≠ 0 is a coupling constant. We show that the critical nonlocal system has a positive ground state solution for negative β and positive large β via variational methods. Moreover, we study the limit behavior of the positive ground state solution ( u β, v β ) as β → − ∞ and some different phenomenon arises comparing with the local Schrödinger system.