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

Abstract

We derive the asymptotic decay of the unique positive, radially symmetric solution to the logarithmic Choquard equation $$-\Delta u+au=\frac{1}{2\pi }[\ln \frac{1}{|x|}\ast |u{|}^2]u\text{in R2}$$ 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).

21 pages