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

Denis Bonheure (1, 2) , Silvia Cingolani (3) , Jean van Schaftingen (4)

1 Département de mathématiques Université Libre de Bruxelles
2 MEPHYSTO - Quantitative methods for stochastic models in physics
3 Polytechnic University of Bari / Politecnico di Bari
4 UCL IRMP - Institut de Recherche en Mathématiques et Physique

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).

Domaines: Equations aux dérivées partielles [math.AP]

Denis Bonheure, Silvia Cingolani, Jean van Schaftingen. The logarithmic Choquard equation : sharp asymptotics and nondegeneracy of the groundstate. 2016. ⟨hal-01414010⟩