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 $I_{\alpha }$ 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 $N$.

18 pages, revised version

Cited by in corpus (1)

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