# American Institute of Mathematical Sciences

March  2019, 39(3): 1497-1515. doi: 10.3934/dcds.2019064

## On finite energy solutions of fractional order equations of the Choquard type

 Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

Received  December 2017 Revised  April 2018 Published  December 2018

Fund Project: The research was supported by NSF of China (No. 11471164, 11871278, 11671209)

Finite energy solutions are the important class of solutions of the Choquard equation. This paper is concerned with the regularity of weak finite energy solutions. For nonlocal fractional-order equations, an integral system involving the Riesz potential and the Bessel potential plays a key role. Applying the regularity lifting lemma to this integral system, we can see that some weak integrable solution has the better regularity properties. In addition, we also show the relation between such an integrable solution and the finite energy solution. Based on these results, we prove that the weak finite energy solution is also the classical solution under some conditions. Finally, we point out that the least energy with the critical exponent can be represented by the sharp constant of some inequality of Sobolev type though the ground state solution cannot be found.

Citation: Yutian Lei. On finite energy solutions of fractional order equations of the Choquard type. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1497-1515. doi: 10.3934/dcds.2019064
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