Uniqueness of positive radial solutions of Choquard type equations

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 $p\in[1,2]$.