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

Abstract

In this article, we study the coupled nonlinear Schrodinger equations with Choquard type nonlinearities $$\displaylines{ -\Delta u+\nu_1u=\mu_1(\frac{1}{|x|^{\alpha}} *u^2)u +\beta (\frac{1}{|x|^{\alpha}} *v^2)u \quad\hbox{in } \mathbb{R}^{N},\cr -\Delta v+\nu_2v=\mu_2(\frac{1}{|x|^{\alpha}} *v^2)v + \beta (\frac{1}{|x|^{\alpha}} *u^2)v \quad\hbox{in } \mathbb{R}^{N},\cr u,v \geq 0\quad \hbox{in } \mathbb{R}^{N}, \quad u,v \in H^{1}(\mathbb{R}^{N}), }$$ where \(\nu_1,\nu_2,\mu_1,\mu_2\) are positive constants, \(\beta>0\) is a coupling constant, \(N\geq 3\), \(\alpha\in(0,N)\cap (0,4)\), and ``*'' is the convolution operator We show that the nonlocal elliptic system has a positive least energy solution for positive small \(\beta\) and positive large \(\beta\) via variational methods. For the case in which \(\nu_1=\nu_2\), \(\mu_1\neq\mu_2\), \(N=3,4,5\) and \(\alpha=N-2\), we prove the uniqueness of positive least energy solutions. Moreover, the asymptotic behaviors of the positive least energy solutions as \(\beta\to 0^{+}\) are studied. For more information see https://ejde.math.txstate.edu/Volumes/2021/47/abstr.html