Existence and asymptotic behavior of positive least energy solutions for coupled nonlinear Choquard equations
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In this article, we study the coupled nonlinear Schrödinger equations with Choquard type nonlinearities
-Δu + v1u = μ1 (1/|x|α ∗ u2)u + β(1/|x|α *v2u in ℝN,
-Δv + v2v = μ2 (1/|x|α ∗ v2)v + β(1/|x|α ∗ u2 in ℝN,
u, v ≥ 0 in ℝN, u, v ∈ H1(ℝN),
where v1, v2, μ1, μ2 are positive constants, β > 0 is a coupling constant, N ≥ 3, α ∈ (0, N) ∩ (0, 4), and "∗" is the convolution operator. We show that the nonlocal elliptic system has a positive least energy solution for positive small β and positive large β via variational methods. For the case in which v1 = v2, energy solutions. Moreover, the asymptotic behaviors of the positive least energy solutions as β → 0+ are studied.
CitationYou, S., Zhao, P., & Wang, Q. (2021). Existence and asymptotic behavior of positive least energy solutions for coupled nonlinear Choquard equations. Electronic Journal of Differential Equations, 2021(47), pp. 1-20.
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