您的位置：山东大学 -> 科技期刊社 -> 《山东大学学报（理学版）》

《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

•   • 上一篇    下一篇

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

RichHTML

3

PDF (PC)

92

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 .
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] 王刚,许信顺*. 一种新的基于多示例学习的场景分类方法[J]. J4, 2010, 45(7): 108 -113 .
[2] 陆玮洁,主沉浮,宋 翠,杨艳丽 . 中药郁金中无机离子的毛细管电泳法测定[J]. J4, 2007, 42(7): 13 -18 .
[3] 唐风琴1,白建明2. 一类带有广义负上限相依索赔额的风险过程大偏差[J]. J4, 2013, 48(1): 100 -106 .
[4] 王　瑶,刘　建,王仁卿,* . 阿利效应及其对生物入侵和自然保护中小种群管理的启示[J]. J4, 2007, 42(1): 76 -82 .
[5] 孙小婷1,靳岚2*. DOSY在寡糖混合物分析中的应用[J]. J4, 2013, 48(1): 43 -45 .
[6] 任敏1,2,张光辉1. 右半直线上依分布收敛独立随机环境中随机游动的吸收概率[J]. J4, 2013, 48(1): 93 -99 .
[7] 徐俊峰. 关于复代数微分方程亚纯解的增长级[J]. J4, 2010, 45(6): 91 -93 .
[8] 谢娟英1, 2，张琰1,谢维信2, 3,高新波2. 一种新的密度加权粗糙K-均值聚类算法[J]. J4, 2010, 45(7): 1 -6 .
[9] 吴志军,沈丹丹. 基于信息综合集成共享的下一代网络化全球航班追踪体系结构及关键技术[J]. 山东大学学报（理学版）, 2016, 51(11): 1 -6 .
[10] 曲守宁,付爱芳,李静,刘静. 基于柔性神经树模型的股票市场风险预测[J]. J4, 2009, 44(11): 44 -47 .
版权所有 © 2018 《山东大学学报(理学版)》编辑部　
地址: 山东省济南市山大南路27号山东大学中心校区（250100）　电话: +86-0531-88366917　E-mail: xblxb@sdu.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发