具有双临界指数的分数阶Kirchhoff方程的正规化解的非存在性结果
A Nonexistence Result of the Normalized Solutions to a Fractional Kirchhoff Equation with Doubly Critical Exponents
摘要: Kirchhoff模型源于研究一根有弹性的绳子在自由振动过程中绳长的改变量，而分数阶Kirchhoff方程则将局部问题延伸到了非局部问题。通过使用变分法，约束极小元思想和一些能量估计，本文证明了一类具有双临界指数和混合非线性项的分数阶Kirchhoff方程的正规化解的非存在性结果，即泛函在一个L2-约束流形上的能量极小元的不存在性。这些能量估计是本文的重点和难点内容，但本文仅针对非存在性结果进行了分析。对分数阶和非局部算子的研究不仅可以应用于数学领域，还能用于连续介质力学，相变现象，博弈论等其他方面。
Abstract: The Kirchhoff model is derived from the study of the length change of an elastic rope during vibration, fractional Kirchhoff equations, however, extend local problems to nonlocal ones. By using the variational method, constrained minimization technique and some energy estimates, a nonexistence result of the normalized solutions to the fractional Kirchhoff equation with doubly critical exponents and combined nonlinearities is obtained in this paper. In other words, the nonexistence of energy minimizers of the functional on the L2-constrained monifold is discussed. These energy estimates are the key and difficult points of this paper. Nevertheless, only the nonexistence result is analyzed here. Besides, the study of fractional order and nonlocal operators can be applied not only to the field of Mathematics, but also to continuum mechanics, phase transition phenomena, game theory, etc.
文章引用：张天晴. 具有双临界指数的分数阶Kirchhoff方程的正规化解的非存在性结果[J]. 应用数学进展, 2024, 13(4): 1817-1826. https://doi.org/10.12677/aam.2024.134171

参考文献

[1] Kirchhoff, G. (1876) Vorlesungen über mathematische Physik. I. Mechanik. Leipzig. Teubner.
[2] Alves, C.O., Corrˆea, 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] Furtado, M.F., de Oliveira, L.D. and da Silva, J.P.P. (2019) Multiple Solutions for a Kirchhoff Equation with Critical Growth. Zeitschrift für angewandte Mathematik und Physik, 70, Article Number 11. [Google Scholar] [CrossRef] 
[4] Li, G. and Ye, H. (2014) Existence of Positive Ground State Solutions for the Nonlinear Kirchhoff Type Equations in. Journal of Differential Equations, 257, 566-600. [Google Scholar] [CrossRef] 
[5] Liu, W. and He, X. (2012) Multiplicity of High Energy Solutions for Superlinear Kirchhoff Equations. Journal of Applied Mathematics and Computing, 39, 473-487. [Google Scholar] [CrossRef] 
[6] Wang, Y., Liu, Y. and Cui, Y. (2018) Multiple Sign-Changing Solutions for Nonlinear Fractional Kirchhoff Equations. Boundary Value Problems, 2018, Article Number: 193. [Google Scholar] [CrossRef] 
[7] Ambrosio, V. (2021) A Multiplicity Result for a Fractional Kirchhoff Equation with a General Nonlinearity. In Nonlinear Fractional Schrödinger Equations in , Birkhäuser, Cham, 363-377. [Google Scholar] [CrossRef] 
[8] Chen, L., Chen, C., Yang, H. and Song, H. (2019) Infinite Radial Solutions for the Fractional Kirchhoff Equation. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 113, 2309-2318. [Google Scholar] [CrossRef] 
[9] Chen, W. and Huang, X. (2022) The Existence of Normalized Solutions for a Fractional Kirchhoff-Type Equation with Doubly Critical Exponents. Zeitschrift für angewandte Mathematik und Physik, 73, Article No. 226. [Google Scholar] [CrossRef] 
[10] Frank, R.L., Lenzmann, E. and Silvestre, L. (2016) Uniqueness of Radial Solutions for the Fractional Laplacian. Communications on Pure and Applied Mathematics, 69, 1671-1726. [Google Scholar] [CrossRef] 
[11] Bisci, G.M., Rãdulescu, V.D. and Servadei, R. (2016) Variational Methods for Nonlocal Fractional Problems. In Encyclopedia of Mathematics, Cambridge University Press, Cambridge.

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