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

Abstract

Consider nonlinear Choquard equations \begin{equation*} \left\{\begin{array}{rcl} -Δu +u & = &(I_α*|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 $I_α$ denotes Riesz potential and $α\in (0, N)$. In this paper, we investigate limit profiles of ground states of nonlinear Choquard equations as $α\to 0$ or $α\to N$. This leads to the uniqueness and nondegeneracy of ground states when $α$ is sufficiently close to $0$ or close to $N$.