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《山东大学学报(理学版)》 ›› 2024, Vol. 59 ›› Issue (8): 48-55,66.doi: 10.6040/j.issn.1671-9352.0.2023.543

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一类带扰动的分数阶临界Choquard方程的正规解

桑彦彬()   

  1. 中北大学数学学院, 山西 太原 030051
  • 收稿日期:2023-12-25 出版日期:2024-08-20 发布日期:2024-07-31
  • 作者简介:桑彦彬(1979—),男,副教授,硕士生导师,博士,研究方向为非线性泛函分析及非线性微分方程. E-mail: sangyanbin@126.com
  • 基金资助:
    山西省基础研究计划资助项目(202103021224198)

Normalized solutions for a class of fractional critical Choquard equations with perturbation

Yanbin SANG()   

  1. School of Mathematics, North University of China, Taiyuan 030051, Shanxi, China
  • Received:2023-12-25 Online:2024-08-20 Published:2024-07-31

摘要:

研究一类分数阶Choquard方程的正规解, 其中非线性项包含Hardy-Littlewood-Sobolev临界指数和带参数的质量超临界非局部项, 分析Pohozaev流形的性质, 建立了上述方程对应能量泛函的Palais-Smale序列的紧性条件。当扰动项的系数充分大时, 获得了其正规基态解的存在性。

关键词: Choquard方程, 正规解, 分数阶算子, 质量超临界, 紧性条件

Abstract:

The normalized solutions for a class of fractional Choquard equations are studied, where Hardy-Littlewood-Sobolev critical exponent and mass supercritical nonlocal term with the parameter are contained in nonlinearites. By analyzing the properties of Pohozaev manifold, the compact condition of Palais-Smale sequences for the energy functional corresponding to above equations is established. When the coefficient of perturbation is large enough, the existence of normalized ground state solutions is obtained.

Key words: Choquard equation, normalized solution, fractional operator, mass supercritical, compact condition

中图分类号: 

  • O175
1 MOLICA BISCI G , RADULESCU V , SERVADEI R . Variational methods for nonlocal fractional problems, with a foreword by Jean Mawhin, encyclopedia of mathematics and its applications[M]. Cambridge: Cambridge University Press, 2016.
2 BHATTARAI S . On fractional Schrödinger systems of Choquard type[J]. Journal of Differential Equations, 2017, 263 (6): 3197- 3229.
doi: 10.1016/j.jde.2017.04.034
3 FENG Binhua , CHEN Ruipeng , REN Jiajia . Existence of stable standing waves for the fractional Schrödinger equations with combined power-type and Choquard-type nonlinearities[J]. Journal of Mathematical Physics, 2019, 60 (5): 051512.
doi: 10.1063/1.5082684
4 LAN Jiali , HE Xiaoming , MENG Yuxi . Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation[J]. Advances in Nonlinear Analysis, 2023, 12 (1): 20230112.
doi: 10.1515/anona-2023-0112
5 YANG Tao . Normalized solutions for the fractional Schrödinger equation with a focusing nonlocal L2-critical or L2-supercritical perturbation[J]. Journal of Mathematical Physics, 2020, 61 (5): 051505.
doi: 10.1063/1.5144695
6 HE X M , RADULESCU V , ZOU W M . Normalized ground states for the critical fractional Choquard equation with a local perturbation[J]. The Journal of Geometric Analysis, 2022, 32 (7): 1- 51.
7 YE Weiwei , SHEN Zifen , YANG Minbo . Normalized solutions for a critical Hartree equation with perturbation[J]. The Journal of Geometric Analysis, 2022, 32 (9): 242.
doi: 10.1007/s12220-022-00986-0
8 CHEN Jianqing , CHEN Zhewen . Normalized ground states for a Hardy-Littlewood-Sobolev upper critical Schrödinger equation with double Choquard type nonlinear terms[J]. Applied Mathematics Letters, 2023, 138, 108521.
doi: 10.1016/j.aml.2022.108521
9 LI Xinfu . Nonexistence, existence and symmetry of normalized ground states to Choquard equations with a local perturbation[J]. Complex Variables and Elliptic Equations, 2023, 68 (4): 578- 602.
doi: 10.1080/17476933.2021.2007378
10 SOAVE N . Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case[J]. Journal of Functional Analysis, 2020, 279 (6): 108610.
doi: 10.1016/j.jfa.2020.108610
11 ALVES C O , CHAO J , MIYAGAKI O H . Normalized solutions for a Schrodinger equation with critical growth in RN[J]. Calculus of Variations and Partial Differential Equations, 2022, 61 (1): 1- 24.
doi: 10.1007/s00526-021-02102-6
12 ZUO J B , RADULESCU V . Normalized solutions to fractional mass supercritical NLS systems with Sobolev critical nonlinearities[J]. Analysis and Mathematical Physics, 2022, 12 (6): 1- 20.
13 DI NEZZA E , PALATUCCI G , VALDINOCI E . Hitchiker's guide to the fractional Sobolev spaces[J]. Bulletin des Sciences Mathématiques, 2012, 136 (5): 521- 573.
doi: 10.1016/j.bulsci.2011.12.004
14 JEANJEAN L . Existence of solutions with prescribed norm for semilinear elliptic equations[J]. Nonlinear Analysis, 1997, 28 (10): 1633- 1659.
doi: 10.1016/S0362-546X(96)00021-1
15 LUO Haijun , ZHANG Zhitao . Normalized solutions to the fractional Schrodinger equations with combined nonlinearities[J]. Calculus of Variations and Partial Differential Equations, 2020, 59 (4): 1- 35.
16 CINGOLANI S , GALLO M , TANAKA K . Symmetric ground states for doubly nonlocal equations with mass constraint[J]. Symmetry, 2021, 13 (7): 1- 17.
[1] 王培婷,李安然,魏重庆. 下临界Choquard型线性耦合系统基态解的存在性[J]. 《山东大学学报(理学版)》, 2019, 54(8): 62-67.
[2] 刘新民,崔玉军* . Banach空间非柱形域上微分系统解的存在性[J]. J4, 2008, 43(4): 1-05 .
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