Mathematische Zeitschrift

, Volume 273, Issue 3–4, pp 883–905 | Cite as

On the regularity of positive solutions of a class of Choquard type equations

  • Yutian Lei


This paper is concerned with positive solutions of a class of Choquard type equations. Such equations are equivalent to integral systems involving the Bessel potential and the Riesz potential. By using two regularity lifting lemmas introduced by Chen and Li [2], we study the regularity for integrable solutions u. We first use the Hardy–Littlewood–Sobolev inequality to obtain an integrability result. Then, it is improved to \({u \in L^s(R^n)}\) for all \({s \in [1, \infty]}\) by an iteration. Next, we use the properties of the contraction map and the shrinking map to prove that u is Lipschitz continuous. Finally, we establish the smoothness of u by a bootstrap argument. Our technique can also be used to handle other integral systems involving the Riesz potential or the Bessel potential, such as the Hartree type equations.


Choquard equation Integral equations Integrability intervals Hardy–Littlewood–Sobolev inequality Bessel potential Regularity lifting lemmas 

Mathematics Subject Classification

35J15 45E10 45G05 


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© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Institute of Mathematics, School of Mathematical SciencesNanjing Normal UniversityNanjingChina

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