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

We derive the asymptotic decay of the unique positive, radially symmetric solution to the logarithmic Choquard equation −∆u + au = 1 2π ln 1 |x| * |u| 2 u in 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)