Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 47, pp. 1-20.
Title: Existence and asymptotic behavior of positive least energy solutions for coupled
nonlinear Choquard equations
Authors: Song You (Tsinghua Univ., Beijing, China)
Peihao Zhao (Lanzhou Univ., Gansu, China)
Qingxuan Wang (Zhejiang Normal Univ., Jinhua, Zhejiang, China)
Abstract:
In this article, we study the coupled nonlinear Schrödinger equations with
Choquard type nonlinearities
$$\displylines{
-\Delta u+\nu_1u=\mu_1(\frac{1}{|x|^{\alpha}} *u^2)u
+\beta (\frac{1}{|x|^{\alpha}} *v^2)u \quad\text{in } \mathbb{R}^{N},\cr
-\Delta v+\nu_2v=\mu_2(\frac{1}{|x|^{\alpha}} *v^2)v
+\beta (\frac{1}{|x|^{\alpha}} *u^2)v \quad\text{in } \mathbb{R}^{N},\cr
u,v \geq 0\quad \text{in } \mathbb{R}^{N}, \quad
u,v \in H^{1}(\mathbb{R}^{N}),
}$$
where ν1,ν2,μ1,μ2
are positive constants, β>0
is a coupling constant, N≥3, α in (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 ν1=ν2, μ1≄μ2,
N=3,4,5 and α=N-2, we prove the uniqueness of positive least energy solutions.
Moreover, the asymptotic behaviors of the positive least energy solutions as
β→ 0+ are studied.
Submitted July 17, 2019. Published May 28, 2021.
Math Subject Classifications: 35B40, 35J47, 35J50.
Key Words: Coupled Choquard equations; positive least energy solution;
asymptotic behavior; variational method.