Positive least energy solutions for coupled nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponent

Abstract

In this paper, we study the existence and nonexistence of positive least energy solutions of the following coupled nonlinear Schrödinger equations with Choquard type nonlinearities: \begin{equation*} \begin{cases} \displaystyle -\Delta u+\nu_{1}u=\mu_{1}\left(\frac{1}{|x|^{4}}\ast u^{2}\right)u +\beta \left(\frac{1}{|x|^{4}}\ast v^{2}\right) u, & x \in \Omega,\\ -\Delta v+\nu_{2}v=\mu_{2}\left(\frac{1}{|x|^{4}}\ast v^{2}\right)v +\beta\left(\frac{1}{|x|^{4}}\ast u^{2}\right)v, & x \in \Omega,\\ u,v \geq 0 \quad\text{in }\Omega, \qquad u=v=0 \quad \text{on } \partial\Omega. \end{cases} \end{equation*} Here $\Omega\subset\mathbb{R}^{N}$ is a smooth bounded domain, $-\lambda_{1}(\Omega)< \nu_{1},\nu_{2}< 0, \lambda_{1}(\Omega)$ is the first eigenvalue of $ (-\Delta, H_{0}^{1}(\Omega))$, $\mu_{1},\mu_{2}> 0$ and $\beta\neq 0$ is a coupling constant. We show that the critical nonlocal elliptic system has a positive least energy solution under appropriate conditions on parameters via variational methods. For the case in which $\nu_{1}=\nu_{2}$, we obtain the classification of the positive least energy solutions. Moreover, the asymptotic behaviors of the positive least energy solutions as $\beta\rightarrow 0$ are studied.