指数型非线性椭圆方程解的渐近行为
Asymptotic Behavior of an Exponential Nonlinear Elliptic Equation Solution
DOI: 10.12677/AAM.2025.142071, PDF,   
作者: 郭锦钰：浙江师范大学数学科学学院，浙江 金华
关键词: 六阶椭圆方程；渐近行为；共形几何；Elliptic Equations of Sixth Order； Asymptotic Behavior； Conformal Geometry
摘要: 本文主要研究的是六阶指数型非线性椭圆方程, 该方程为负的拉普拉斯算子的三次方作用于 u(x) 等于 e 的 u(x) 次方. e 的 u(x) 次方函数在 R 的六维空间去掉一个单位球 B 的区域上是勒贝格 可积的, 其中单位球 B 是由 R 六维空间中满足 x 的模小于 1 的所有 x 组成. 当 x 的模趋于无穷 大的时候, u(x)/ ln |x| 的极限是 α, 其中 α 小于 -6.
Abstract: This paper mainly studies the sixth-order exponential nonlinear elliptic equation, which is the cube of the negative Laplacian acting on u equal to e to the power of u. The function e to the power of u(x) is Lebesgue integrable in the region of the six-dimensional space of R excluding a unit ball B, where the unit ball B is composed of all x in the six-dimensional space of R satisfying the modulus of x is less than 1. When the modulus of x tends to infinity, the limit of u(x)/ ln |x| is α, where α is less than -6.
文章引用：郭锦钰. 指数型非线性椭圆方程解的渐近行为[J]. 应用数学进展, 2025, 14(2): 286-301. https://doi.org/10.12677/AAM.2025.142071

参考文献

[1] Graham, C.R., Jenne, R., Mason, L.J. and Sparling, G.A.J. (1992) Conformally Invariant Powers of the Laplacian, I: Existence. Journal of the London Mathematical Society, 2, 557- 565.
https://doi.org/10.1112/jlms/s2-46.3.557
[2] Chang, S.A. and Yang, P.C. (1997) On Uniqueness of Solutions of n-th Order Differential Equations in Conformal Geometry. Mathematical Research Letters, 4, 91-102.
https://doi.org/10.4310/mrl.1997.v4.n1.a9
[3] Graham, C.R., Jenne, R., Mason, L.J. and Sparling, G.A.J. (1992) Conformally Invariant Powers of the Laplacian, I: Existence. Journal of the London Mathematical Society, 2, 557- 565.
https://doi.org/10.1112/jlms/s2-46.3.557
[4] Chen, W. and Li, C. (1991) Classification of Solutions of Some Nonlinear Elliptic Equations. Duke Mathematical Journal, 63, 615-622.
https://doi.org/10.1215/s0012-7094-91-06325-8
[5] Lin, C.-S. (1998) A Classification of Solutions of a Conformally Invariant Fourth Order Equa- tion in Rn. Commentarii Mathematici Helvetici, 73, 206-231.
https://doi.org/10.1007/s000140050052
[6] Martinazzi, L. (2008) Classification of Solutions to the Higher Order Liouville’s Equation on R2m. Mathematische Zeitschrift, 263, 307-329.
https://doi.org/10.1007/s00209-008-0419-1
[7] Wei, J. and Ye, D. (2008) Nonradial Solutions for a Conformally Invariant Fourth Order Equation in R4. Calculus of Variations and Partial Differential Equations, 32, 373-386.
https://doi.org/10.1007/s00526-007-0145-2
[8] Wei, J. and Xu, X. (1999) Classification of Solutions of Higher Order Conformally Invariant Equations. Mathematische Annalen, 313, 207-228.
https://doi.org/10.1007/s002080050258
[9] Goldberg, S.I. (1977) A Uniqueness Theorem for Surfaces in the Large. Hokkaido Mathematical Journal, 6, 28-30.
[10] Guo, Z., Huang, X. and Zhou, F. (2015) Radial Symmetry of Entire Solutions of a Bi-Harmonic Equation with Exponential Nonlinearity. Journal of Functional Analysis, 268, 1972-2004.
https://doi.org/10.1016/j.jfa.2014.12.010

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