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

Yutian Lei

doi:10.3934/dcds.2019064

Article Contents

2019, Volume 39, Issue 3: 1497-1515. Doi: 10.3934/dcds.2019064

This issue Previous Article A Hopf's lemma and the boundary regularity for the fractional p-Laplacian Next Article A priori bounds and existence result of positive solutions for fractional Laplacian systems

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

Yutian Lei

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

Received: November 30, 2017

Revised: March 31, 2018

Published: December 2018

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

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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.

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Mathematics Subject Classification: 35J91, 35Q55, 45E10, 45G05.

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