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

We derive the asymptotic decay of the unique positive, radially symmetric solution to the logarithmic Choquard equation -Delta u + au = 1/2 pi [ln 1/vertical bar x vertical bar * vertical bar u vertical bar(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 (2009) [13]. (C) 2017 Elsevier Inc. All rights reserved.