Nonlinear Choquard equations involving nonlocal operators

In this paper, we study nonlinear Choquard equations \begin{equation}\label{eq 1a1-} (-\Delta+id)^{\frac{1}{2}}u=(I_\alpha*{|u|^p})|u|^{p-2}u\ \ {\rm in} \ \ \mathbb{R}^N, \ \ \ u\in H^{\frac{1}{2}}(\mathbb{R}^N), \end{equation} where $(-\Delta+id)^\frac{1}{2}$ is a nonlocal operator, $p>0$, $N\geq2$ and $I_\alpha$ is the Riesz potential with order $\alpha\in(0,N)$. We show that there is a ground state solution to the above problem if $\frac{N+\alpha}{N}<p<\frac{N+\alpha}{N-1}$ and no solution if $0<p\leq\frac{N+\alpha}{N+1}$ or $p\geq\frac{N+\alpha}{N-1}$. Furthermore, the existence of infinity many solutions to the above problem is discussed when $p$ satisfies that $\frac{N+\alpha}{N}<p<\frac{N+\alpha}{N-1}$.