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| 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 |
| 投稿时间：2024-12-12 修订日期：2025-03-08 |
| DOI： |
| 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］YB23）；2025年贵州省基础研究计划项目：黔科合基础-zk[2025]面上218 |
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| 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,我们首先构造了一个能量泛函,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.另外,我们还研究了山路解的存在性.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: |
| In this paper, we study the existence of normalized solutions for the Choquard equation with harmonic The Choquard equation with harmonic potential and general nonlinear terms is an important class of nonlinear partial differential equation, 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 minimizers and mountain pass solutions. Using the variational method, we first construct an energy functional. By analyzing the properties of this energy functional, especially its minimization property in Sobolev spaces, and applying the Palais-Smale condition, we further ensure the existence and smoothness of local minimizers in appropriate Sobolev spaces through the analysis of the influence of the nonlinear terms and harmonic potential. Additionally, we study the existence of mountain pass solution, which 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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