Existence and asymptotic behavior of positive least energy solutions for coupled nonlinear Choquard equations

Abstract

<p>In this article, we study the coupled nonlinear Schrödinger equations with Choquard type nonlinearities</p> <pre>-Δu + v₁u = μ₁ (1/|x|^α ∗ u<sup</sup>2)u + β(1/|x|^α *v²u in ℝ^N,<br> -Δv + v₂v = μ₂ (1/|x|^α ∗ v²)v + β(1/|x|^α ∗ u² in ℝ^N,<br> u, v ≥ 0 in ℝ^N, u, v ∈ H¹(ℝ^N),</pre> <p>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.</p>