  In this paper, we consider the following Choquard type equation
\begin{equation} \left\{\begin{aligned} &-\Delta u+\lambda
u=\gamma(\Phi_N(|x|)\ast|u|^p)u \ \ \mbox{in $\mathbb{R}^N$}, \\
&\lim\limits_{|x|\to\infty}u(x)=0,\\ \end{aligned}\right. \end{equation}
  where $N\geq2,\lambda>0,\gamma>0, p\in[1,2]$ and $\Phi_N(|x|)$ denotes the
fundamental solution of the Laplacian $-\Delta$ on $\mathbb{R}^N$. This
equation does not have a variational frame when $p\neq 2.$ Instead of
variational methods, we prove the existence and uniqueness of positive radial
solutions of the above equation via the shooting method by establishing some
new differential inequalities. The proofs are based on an analysis of the
corresponding system of second-order differential equations, and our results
extend the existing ones in the literature from $p=2$ to $p\in[1,2]$.
