| 樊再美，张家锋*.含有调和位势和一般非线性项的Choquard方程
正规化解的存在性[J].海南师范大学学报自科版,2025,38(3):264-273 |
| 含有调和位势和一般非线性项的Choquard方程
正规化解的存在性 |
| The Existence of Normalized Solutions for the Choquard Equation with Harmonic Potential and General Nonlinear Terms |
| |
| DOI：10.12051/j.issn.1674-4942.2025.03.002 |
| 中文关键词: Choquard方程 调和位势 约束变分法 正规化解 |
| 英文关键词: Choquard equation harmonic potential constrained variational method normalized solution |
| 基金项目:国 家 自 然 科 学 基 金 项 目（11861021）；贵 州 省 教 育 厅 自 然 科 学 研 究 项 目（QJJ2023012，QJJ2023061，
QJJ2023062）；贵州民族大学自然科学研究项目（GZMUZK［2022］YB06）；贵州省基础研究计划项目（黔科合基础 MS
［2025］218） |
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| 全文下载次数: 155 |
| 中文摘要: |
| 含有调和位势和一般非线性项的Choquard方程是一类重要的非线性偏微分方程，广泛
应用于量子力学和物理学中。本文研究了该类方程正规化解的存在性，证明了局部极小解和山路
解的存在性。通过变分法，首先构造了 1 个能量泛函，通过分析该能量泛函的性质，尤其是在
Sobolev空间中的极小性质，并结合Palais-Smale条件，通过对非线性项和调和位势影响进行分析，
进一步确保了局部极小解在适当的Sobolev空间中是存在的且是光滑的；由于山路解的出现与能量
泛函的拓扑性质密切相关，通过深入研究能量泛函的拓扑结构，并运用变分方法中的精细技术，通
过分析非线性项和调和位势的特定性质，进一步证明了山路解的存在性，从而证明了方程多个解
的存在性。本文的研究结果不仅在数学上具有重要意义，也为相关物理问题的深入分析提供了理
论支持，并推广了已有文献中的成果。 |
| 英文摘要: |
| The Choquard equation with harmonic potential and general nonlinear terms is an important class of nonlinear
partial differential equation and widely applied in quantum mechanics and physics. This paper investigates the existence of
normalized solutions for this type of equation, and prove the existence of local minima and mountain pass solutions. Using
the variational method, we first construct an energy functional. By analyzing the properties of this energy functional, espe⁃
cially its minimization property in Sobolev spaces, and applying the Palais-Smale condition, we further ensure the exis⁃
tence and smoothness of local minima in appropriate Sobolev spaces through the analysis of the influence of the nonlinear
terms and harmonic potential. The existence of mountain pass solution is closely related to the topological properties of the
energy functional. By delving into the topological structure of the energy functional and applying refined techniques from
variational methods, we further prove the existence of mountain pass solution through the analysis of the specific properties of the
nonlinear terms and harmonic potential. This provides the existence of multiple solutions to the equation. The results of this
paper are not only of significant mathematical importance but also provide theoretical support for the in-depth analysis of related physical problems, extending the results in the existing literature. |
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