Positive ground state solutions for a class of fractional coupled Choquard systems

In this paper, we combine the critical point theory and variational method to investigate the following a class of coupled fractional systems of Choquard type

$$\{\begin{array}{llll}(-\mathrm{\Delta }{)}^{s}u+{\lambda }_{1}u& =(I_{\alpha }\ast |u{|}^p)|u{|}^{p-2}u+\beta v& & \text{in}\\ {\mathbb{R}}^N,(-\mathrm{\Delta }{)}^{s}v+{\lambda }_{2}v& =(I_{\alpha }\ast |v{|}^p)|v{|}^{p-2}v+\beta u& & \text{in}{\mathbb{R}}^N,\end{array}$$

with $s\in (0,1),N\ge 3,\alpha \in (0,N),p>1$, ${\lambda }_i>0$ are constants for $i=1,2$, $\beta >0$ is a parameter, and $I_{\alpha }(x)$ is the Riesz Potential. We prove the existence and asymptotic behaviour of positive ground state solutions of the systems by using constrained minimization method and Hardy-Littlewood-Sobolev inequality. Moreover, nonexistence of nontrivial solutions is also obtained.