Limit profiles and uniqueness of ground states to the nonlinear Choquard equations

Consider nonlinear Choquard equations \begin{equation*} \left{\begin{array}{rcl} -\Delta u +u & = &(I_\alpha*|u|^p)|u|^{p-2}u \quad $\text{in }$ $\mathbb{R}$^N, \ \lim_{x \to \infty}u(x) & = &0, \end{array}\right. \end{equation*} where $I_{\alpha }$ denotes Riesz potential and $\alpha \in (0, N)$. In this paper, we investigate limit profiles of ground states of nonlinear Choquard equations as $\alpha \to 0$ or $\alpha \to N$. This leads to the uniqueness and nondegeneracy of ground states when $\alpha$ is sufficiently close to $0$ or close to $N$.