Regularity and classification of solutions to static Hartree equations involving fractional Laplacians

Wei Dai; Jiahui Huang; Yu Qin; Bo Wang; Yanqin Fang

doi:10.3934/dcds.2018117

Article Contents

2019, Volume 39, Issue 3: 1389-1403. Doi: 10.3934/dcds.2018117

This issue Previous Article Non-existence of positive solutions for a higher order fractional equation Next Article Regularity estimates for nonlocal Schrödinger equations

Regularity and classification of solutions to static Hartree equations involving fractional Laplacians

1.
School of Mathematics and Systems Science, Beihang University (BUAA), Beijing 100083, China
2.
School of Mathematics, Hunan University, Changsha 410082, China
3.
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, 2522, NSW, Australia

* Corresponding author: Wei Dai at weidai@buaa.edu.cn
* Corresponding author: Wei Dai at weidai@buaa.edu.cn

Received: April 30, 2017

Revised: October 31, 2017

Published: December 2018

The first author was supported by the NNSF of China (No. 11501021), the second author was supported by the NNSF of China (No. 11301166)

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Abstract

In this paper, we are concerned with the fractional order equations (1) with Hartree type $ \dot{H}^{\frac{α}{2}} $-critical nonlinearity and its equivalent integral equations (3). We first prove a regularity result which indicates that weak solutions are smooth (Theorem 1.2). Then, by applying the method of moving planes in integral forms, we prove that positive solutions $ u $ to (1) and (3) are radially symmetric about some point $ x_{0}∈\mathbb{R}^{d} $ and derive the explicit forms for $ u $ (Theorem 1.3 and Corollary 1). As a consequence, we also derive the best constants and extremal functions in the corresponding Hardy-Littlewood-Sobolev inequalities (Corollary 2).

Keywords:

Fractional Laplacians,
positive solutions,
radial symmetry,
uniqueness,
regularity,
Hartree type nonlinearity,
methods of moving planes in integral forms.

Mathematics Subject Classification: Primary: 35R11, 35J91; Secondary: 35B06, 35B65.

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