Ground states of coupled critical Choquard equations with weighted potentials

In this paper, we are concerned with the following coupled Choquard type system with weighted potentials \[\begin{cases} -\Delta u+V_{1}(x)u=\mu_{1}(I_{\alpha}\!\ast\![Q(x)|u|^{\frac{N+\alpha}{N}}])Q(x)|u|^{\frac{\alpha}{N}-1}u+\beta(I_{\alpha}\!\ast\![Q(x)|v|^{\frac{N+\alpha}{N}}])Q(x)|u|^{\frac{\alpha}{N}-1}u,\\ -\Delta v+V_{2}(x)v=\mu_{2}(I_{\alpha}\!\ast\![Q(x)|v|^{\frac{N+\alpha}{N}}])Q(x)|v|^{\frac{\alpha}{N}-1}v+\beta(I_{\alpha}\!\ast\![Q(x)|u|^{\frac{N+\alpha}{N}}])Q(x)|v|^{\frac{\alpha}{N}-1}v,\\ u,v\in H^{1}(\mathbb{R}^{N}),\end{cases}\] where \(N\geq3\), \(\mu_{1},\mu_{2},\beta\gt 0\) and \(V_{1}(x)\), \(V_{2}(x)\) are nonnegative functions. Via the variational approach, one positive ground state solution of this system is obtained under some certain assumptions on \(V_{1}(x)\), \(V_{2}(x)\) and \(Q(x)\). Moreover, by using Hardy's inequality and one Pohozǎev identity, a non-existence result of non-trivial solutions is also considered.