The logarithmic Choquard equation: sharp asymptotics and nondegeneracy of the groundstate

TL;DR

This paper analyzes the decay behavior and nondegeneracy of the unique positive radially symmetric solution to the logarithmic Choquard equation in two dimensions, extending understanding of its mathematical properties and stability.

Contribution

It provides the first detailed asymptotic decay analysis and proves the nondegeneracy of the ground state for the 2D logarithmic Choquard equation, building on prior 3D results.

Findings

Derived sharp asymptotic decay of the ground state

Established nondegeneracy of the solution

Extended nondegeneracy results to 2D case

Abstract

We derive the asymptotic decay of the unique positive, radially symmetric solution to the logarithmic Choquard equation $$ - \Delta u + a u = \frac{1}{2 \pi} \Bigl[\ln \frac{1}{|x|}* |u|^2 \Bigr] \ u \qquad \text{in $\mathbb{R}^2$} $$ and we establish its nondegeneracy. For the corresponding three-dimensional problem, the nondegeneracy property of the positive ground state to the Choquard equation was proved by E. Lenzmann (Analysis & PDE, 2009).