带对数非线性项的Kirchhoff方程的规范解
Normalized Solutions for Kirchhoff Equations with Logarithmic Nonlinearity
DOI: 10.12677/aam.2026.154138, PDF,    科研立项经费支持
作者: 陆瑞康, 谢启林*：广东工业大学数学与统计学院，广东 广州
关键词: 规范解；基尔霍夫方程；对数非线性项；Normalized Solutions； Kirchhoff Equation； Logarithmic Nonlinearity
摘要: 本文旨在研究一类带有对数非线性项的Kirchhoff型方程规范解的存在性问题。文章首先分析了与方程对应的能量泛函在不同参数范围下的几何结构，随后运用变分方法中的极小化序列技巧，结合集中紧性原理研究了该方程是否存在规范解。主要结果表明：在质量次临界和质量临界情形下，该方程存在一个全局基态解；而在质量超临界情形中，则存在一个局部基态解。对比目前现有的结果，本文的结论是对现有相关结果的推广。
Abstract: This paper aims to investigate the existence of normalized solutions to a class of Kirchhoff-type equations with logarithmic nonlinearities. The paper first analyzes the geometric structure of the corresponding energy functional under different parameter regimes. Subsequently, variational methods, specifically the technique of minimizing sequences, are applied in conjunction with the concentration compactness principle to systematically explore whether normalized solutions exist for the equation. The main results show that in mass subcritical and mass critical cases, the equation admits a global ground state solution; whereas in the mass supercritical case, a local ground state solution exists. Compared to existing results, the conclusions of this paper extend the current findings in the field.
文章引用：陆瑞康, 谢启林. 带对数非线性项的Kirchhoff方程的规范解[J]. 应用数学进展, 2026, 15(4): 78-86. https://doi.org/10.12677/aam.2026.154138

参考文献

[1] Lions, J.L. (1978) On Some Questions in Boundary Value Problems of Mathematical Physics. In: North-Holland Mathematics Studies, Elsevier, 284-346. [Google Scholar] [CrossRef] 
[2] Alves, C.O., Corrêa, F.J.S.A. and Ma, T.F. (2005) Positive Solutions for a Quasilinear Elliptic Equation of Kirchhoff Type. Computers & Mathematics with Applications, 49, 85-93. [Google Scholar] [CrossRef] 
[3] Lei, C.Y., Liao, J.F. and Tang, C.L. (2015) Multiple Positive Solutions for Kirchhoff Type of Problems with Singularity and Critical Exponents. Journal of Mathematical Analysis and Applications, 421, 521-538. [Google Scholar] [CrossRef] 
[4] Zloshchastiev, K.G. (2010) Logarithmic Nonlinearity in Theories of Quantum Gravity: Origin of Time and Observational Consequences. Gravitation and Cosmology, 16, 288-297. [Google Scholar] [CrossRef] 
[5] Shuai, W. and Yang, X.L. (2023) Normalized Solutions for Logarithmic Schrödinger Equation with a Perturbation of Power Law Nonlinearity.
[6] Li, G.B., Luo, X. and Yang, T. (2022) Normalized Solutions to a Class of Kirchhoff Equations with Sobolev Critical Exponent. Annales Fennici Mathematici, 47, 895-925. [Google Scholar] [CrossRef] 
[7] Cazenave, T. (1983) Stable Solutions of the Logarithmic Schrödinger Equation. Nonlinear Analysis: Theory, Methods & Applications, 7, 1127-1140. [Google Scholar] [CrossRef] 
[8] Stuart, C.A. (1981) Bifurcation from the Continuous Spectrum in the L2 Theory of Elliptic Equations on RN. In: Recent Methods in Nonlinear Analysis and Applications, Liguori, 231-300.
[9] Weinstein, M.I. (1983) Nonlinear Schrödinger Equations and Sharp Interpolation Estimates. Communications in Mathematical Physics, 87, 567-576. [Google Scholar] [CrossRef] 
[10] Bartsch, T. and Soave, N. (2017) A Natural Constraint Approach to Normalized Solutions of Nonlinear Schrödinger Equations and Systems. Journal of Functional Analysis, 272, 4998-5037. [Google Scholar] [CrossRef] 
[11] Lieb, E.H. and Loss, M. (2001) Analysis Graduate Studies in Mathematics Vol 14. American Mathematical Society.
[12] Vázquez, J.L. (1984) A Strong Maximum Principle for Some Quasilinear Elliptic Equations. Applied Mathematics & Optimization, 12, 191-202. [Google Scholar] [CrossRef] 
[13] Mederski, J. and Schino, J. (2022) Least Energy Solutions to a Cooperative System of Schrödinger Equations with Prescribed L2-Bounds: At Least L2-Critical Growth. Calculus of Variations and Partial Differential Equations, 61, Article 10.

为你推荐