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

TL;DR

This paper studies the behavior of ground states in nonlinear Choquard equations as the parameter alpha approaches 0 or N, establishing their limit profiles, uniqueness, and nondegeneracy in these regimes.

Contribution

It provides the first analysis of limit profiles and proves uniqueness and nondegeneracy of ground states for extreme alpha values in nonlinear Choquard equations.

Findings

Limit profiles of ground states as alpha approaches 0 or N are characterized.

Uniqueness of ground states is established for alpha near 0 or N.

Nondegeneracy of ground states is proved in these regimes.

Abstract

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 $N$.