Ground state solutions of a class of critical Choquard systems with logarithmic terms

SANG Yanbin; YU Yan; SHI Na

doi:10.13471/j.cnki.acta.snus.ZR20240032

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Ground state solutions of a class of critical Choquard systems with logarithmic terms

Research articles | 更新时间：2025-11-14

- Ground state solutions of a class of critical Choquard systems with logarithmic terms
- Acta Scientiarum Naturalium Universitatis Sunyatseni Vol. 64, Issue 2, Pages: 138-147(2025)
- 作者机构：
  
  中北大学数学学院，山西太原 030051
- 作者简介：
- 基金信息：
- DOI：10.13471/j.cnki.acta.snus.ZR20240032
  CLC： O175.8
- Received：21 January 2024，
  
  Accepted：28 May 2024，
  
  Published Online：25 July 2024，
  
  Published：15 March 2025
- 稿件说明：
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桑彦彬,蔚艳,史娜.一类带有对数项的临界Choquard方程组的基态解[J].中山大学学报(自然科学版)(中英文),2025,64(02):138-147.

SANG Yanbin,YU Yan,SHI Na.Ground state solutions of a class of critical Choquard systems with logarithmic terms[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2025,64(02):138-147.
桑彦彬,蔚艳,史娜.一类带有对数项的临界Choquard方程组的基态解[J].中山大学学报(自然科学版)(中英文),2025,64(02):138-147. DOI： 10.13471/j.cnki.acta.snus.ZR20240032.

SANG Yanbin,YU Yan,SHI Na.Ground state solutions of a class of critical Choquard systems with logarithmic terms[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2025,64(02):138-147. DOI： 10.13471/j.cnki.acta.snus.ZR20240032.

摘要

考虑一类Choquard型耦合方程组，其中非线性项含有对数项和Hardy-Littlewood-Sobolev临界指数. 当对数项的系数均为负值时，借助单个临界Choquard方程相应的局部极小点的存在性，建立了该系统对应能量泛函在Nehari流形中Palais-Smale序列的收敛性，进而利用Ekeland变分原理，获得了其具有极小能量的正解存在性. 同时在对参数施加与线性问题相关的第一特征值的限制条件下，构造了上述系统具有负能量水平的非负解的存在性. 本文的结果扩展了对数项系数为正值的情形，分析了系数的负性对能量泛函几何结构的影响，是对经典Sobolev临界系统在Choquard算子上的推广和延伸.

Abstract

A class of Choquard type coupled systems are considered， where Hardy-Littlewood-Sobolev critical exponents and logarithmic terms are contained in nonlinear terms. If the coefficients of logarithmic terms are both negative， Palais-Smale sequences of the energy functional corresponding to above problems in Nehari manifold are established by using of the existence on a local minima of single critical Choquard equation. Furthermore， by adopting Ekeland’s variational principle， some restricted conditions under which the parameters are related to the first eigenvalue of linear operator with Dirichlet boundary conditions are given. The nonnegative solution with negative energy level of above systems is obtained. Our work generalizes the cases that the coefficients of logarithmic terms are positive， and analyzes the impact of negative coefficients on geometry structure of the energy functional. In fact， our results extend classical Sobolev critical systems to the corresponding Choquard problems.

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references

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