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).
Categories: math.AP