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arXiv · 1612.02194

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

Denis Bonheure·Silvia Cingolani·Jean Van Schaftingen
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Abstract

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).

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Denis Bonheure, Silvia Cingolani, Jean Van Schaftingen. 2016-12-07. The logarithmic Choquard equation: sharp asymptotics and nondegeneracy of the groundstate. https://doi.org/10.1016/j.jfa.2017.02.026

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