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Existence of groundstates for a class of nonlinear Choquard equations


Authors: Vitaly Moroz and Jean Van Schaftingen
Journal: Trans. Amer. Math. Soc. 367 (2015), 6557-6579
MSC (2010): Primary 35J61; Secondary 35B33, 35B38, 35B65, 35Q55, 45K05
DOI: https://doi.org/10.1090/S0002-9947-2014-06289-2
Published electronically: December 18, 2014
MathSciNet review: 3356947
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Abstract: We prove the existence of a nontrivial solution $ u \in H^1 (\mathbb{R}^N)$ to the nonlinear Choquard equation

$\displaystyle - \Delta u + u = \bigl (I_\alpha \ast F (u)\bigr ) F' (u)$$\displaystyle \quad \text {in \(\mathbb{R}^N\),} $

where $ I_\alpha $ is a Riesz potential, under almost necessary conditions on the nonlinearity $ F$ in the spirit of Berestycki and Lions. This solution is a groundstate and has additional local regularity properties; if moreover $ F$ is even and monotone on $ (0,\infty )$, then $ u$ is of constant sign and radially symmetric.

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Additional Information

Vitaly Moroz
Affiliation: Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, Wales, United Kingdom
Email: V.Moroz@swansea.ac.uk

Jean Van Schaftingen
Affiliation: Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, Chemin du Cyclotron 2 bte L7.01.01, 1348 Louvain-la-Neuve, Belgium
Email: Jean.VanSchaftingen@uclouvain.be

DOI: https://doi.org/10.1090/S0002-9947-2014-06289-2
Keywords: Stationary Choquard equation, stationary nonlinear Schr\"odinger--Newton equation, stationary Hartree equation, Riesz potential, nonlocal semilinear elliptic problem, Poho\v{z}aev identity, existence, variational method, groundstate, mountain pass, symmetry, polarization
Received by editor(s): March 14, 2013
Received by editor(s) in revised form: September 22, 2013
Published electronically: December 18, 2014
Additional Notes: The second author was supported by the Grant n. 2.4550.10 “Étude qualitative des solutions d’équations aux dérivées partielles elliptiques” of the Fonds de la Recherche Fondatementale Collective (Fédération Wallonie–Bruxelles).
Article copyright: © Copyright 2014 American Mathematical Society
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