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

Bonheure, Denis [Université Libre de Bruxelles] Cingolani, Silvia [Politecnico di Bari] Van Schaftingen, Jean [UCL]

We derive the asymptotic decay of the unique positive, radially symmetric solution to the logarithmic Choquard equation −Δu+au=1/(2π) [ln1|x|∗|u|²] u 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).