Existence of groundstates for a class of nonlinear Choquard equations
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- by Vitaly Moroz and Jean Van Schaftingen PDF
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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 \[ - \Delta u + u = \bigl (I_\alpha \ast F (u)\bigr ) F’ (u) \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.References
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Additional Information
- Vitaly Moroz
- Affiliation: Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, Wales, United Kingdom
- MR Author ID: 359396
- 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
- MR Author ID: 730276
- ORCID: 0000-0002-5797-9358
- Email: Jean.VanSchaftingen@uclouvain.be
- 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).
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3356947