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−Δu+au=12π[ln⁡1|x|⁎|u|2]uin 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 (2009) [13].