Dirichlet 形式的 Bahri-Lions 型定理及其在非线性退化椭圆方程中的应用——献给陈化教授 70 寿辰

数学物理学报 ›› 2026, Vol. 46 ›› Issue (2): 428-451.

Dirichlet 形式的 Bahri-Lions 型定理及其在非线性退化椭圆方程中的应用——献给陈化教授 70 寿辰

樊云露¹(), 廖昕²^,^*()

¹ 武汉大学数学与统计学院武汉 430072
² 湖南师范大学大学数学与统计学院长沙 410081

收稿日期:2025-10-31 修回日期:2025-12-23 出版日期:2026-04-26 发布日期:2026-04-27
通讯作者: 廖昕 E-mail:yunlufan@whu.edu.cn;xin_liao@whu.edu.cn
作者简介:樊云露, Email：yunlufan@whu.edu.cn
基金资助:
国家自然科学基金(12571249)

A Bahri-Lions Type Theorem for Dirichlet Forms and Its Applications to Nonlinear Degenerate Elliptic Equations

Yunlu Fan¹(), Xin Liao²^,^*()

¹ School of Mathematics and Statistics, Wuhan University, Wuhan 430072
² School of Mathematics and Statistics, Hunan Normal University, Changsha 410081

Received:2025-10-31 Revised:2025-12-23 Online:2026-04-26 Published:2026-04-27
Contact: Xin Liao E-mail:yunlufan@whu.edu.cn;xin_liao@whu.edu.cn
Supported by:
NSFC(12571249)

摘要/Abstract

摘要：

该文将 1988 年 Bahri-Lions 的结果推广至与 Dirichlet 形式相关的半线性方程, 通过相对亏格构造了一类新的极小极大结构, 并给出对应的 Morse 指标估计. 该类问题在几何分析, 椭圆与退化椭圆方程研究中具有重要意义.

关键词: Dirichlet 形式, 半线性退化椭圆方程, Morse 指标, 变号解

Abstract:

In this paper, we extend the Bahri-Lions theorem (1988) to a class of semilinear problems associated with Dirichlet forms. By introducing a new min-max scheme based on the notion of relative genus, we construct novel critical point structures and establish corresponding estimates for the Morse index of the obtained solutions. The results provide a unified framework for treating variational problems arising from degenerate and non-uniformly elliptic equations, and are expected to have further applications in geometric analysis and the study of elliptic and degenerate elliptic partial differential equations.

Key words: Dirichlet forms, semilinear degenerate elliptic equations, Morse index, sign-changing solutions

中图分类号:

O175.25

樊云露, 廖昕. Dirichlet 形式的 Bahri-Lions 型定理及其在非线性退化椭圆方程中的应用——献给陈化教授 70 寿辰[J]. 数学物理学报, 2026, 46(2): 428-451.

Yunlu Fan, Xin Liao. A Bahri-Lions Type Theorem for Dirichlet Forms and Its Applications to Nonlinear Degenerate Elliptic Equations[J]. Acta mathematica scientia,Series A, 2026, 46(2): 428-451.

参考文献 32

[1]	Bahri A, Lions P L. Solutions of superlinear elliptic equations and their Morse indices. Comm Pure Appl Math, 1992, 45(9): 1205-1215
[2]	Barhi A, Lions P L. Morse index of some min-max critical points. I. Application to multiplicity results, Comm Pure Appl Math, 1988, 41(8): 1027-1037
[3]	Bartsch T, Liu Z, Weth T. Sign changing solutions of superlinear Schrödinger equations. Comm Partial Differ Equ, 2004, 29(1/2): 25-42
[4]	Bonfiglioli A, Lanconelli E, Uguzzoni F. Stratified Lie Groups and Potential Theory for Their Sub-Laplacians. Berlin: Springer, 2007
[5]	Chang K C. Methods in Nonlinear Analysis. Berlin: Springer, 2005
[6]	Chang K C. Infinite Dimension Morse Theory and Multiple Solution Problems. Boston: Springer, 1993
[7]	Chen H, Chen H G. Estimates of eigenvalues for subelliptic operators on compact manifold. J Math Pures Appl, 2019, 131: 64-87
[8]	Chen H, Chen H G. Estimates of Dirichlet eigenvalues for a class of sub-elliptic operators. Proc Lond Math Soc, 2021, 122(6): 808-847
[9]	Chen H, Chen H G, Yuan X R. Existence and multiplicity of solutions to Dirichlet problem for semilinear subelliptic equation with a free perturbation. J Differ Equ, 2022, 341: 504-537
[10]	Chen H, Chen H G, Li J N, Liao X. Multiplicity of sign-changing solutions for semilinear subelliptic Dirichlet problem. arXiv:2510.10120
[11]	Coffman C V. Lyusternik-Schnirelman theory: complementary principles and the Morse index. Nonlinear Anal, 1988, 12(5): 507-529
[12]	Davies E B. Heat Kernels and Spectral Theory. Cambridge: Cambridge University Press, 1989
[13]	Dugundji J. An extension of Tietze's theorem. Pac J Math, 1951, 1: 353-367
[14]	Frank R L, Lieb E H, Seiringer R. Equivalence of Sobolev inequalities and Lieb-Thirring inequalities. XVIth International Congress on Mathematical Physics, 2010, 2015: 523-535
[15]	Fukushima M, Oshima Y, Takeda M. Dirichlet Forms and Symmetric Markov Processes. Berlin: De Gruyter, 2010
[16]	Ghoussoub N. Duality and Perturbation Methods in Critical Point Theory. Cambridge: Cambridge University Press, 1993
[17]	Ghosh S, Kumar V, Ruzhansky M. Compact embeddings, eigenvalue problems, and subelliptic Brezis-Nirenberg equations involving singularity on stratified Lie groups. Math Ann, 2024, 388(4): 4201-4249
[18]	Hörmander L. Hypoelliptic second order differential equations. Acta Math, 1967, 119(1): 147-171
[19]	Kogoj A E, Lanconelli E. On semilinear $\triangle_\lambda$-Laplace equation. Nonlinear Analysis-Theory Methods and Applications, 2012, 75(12): 4637-4649
[20]	Lazer A C, Solimini S. Nontrivial solutions of operator equations and Morse indices of critical points of min-max type. Nonlinear Anal, 1988, 12(8): 761-775
[21]	Levin D, Solomyak M. The Rozenblum-Lieb-Cwikel inequality for Markov generators. J Anal Math, 1997, 71: 173-193
[22]	Marino A A, Prodi G. Metodi perturbativi nella teoria di Morse. Boll Un Mat Ital, 1975, 11(3): 1-32
[23]	Matzeu M. Mountain pass and linking type solutions for semilinear Dirichlet forms. Recent Trends in Nonlinear Analysis, 2000, 40: 217-231
[24]	Rabinowitz P H. Minimax Methods in Critical Point Theory with Applications to Differential Equations. Providence, RI: Amer Math Soc, 1986
[25]	Rabinowitz P H. Multiple critical points of perturbed symmetric functionals. Trans Amer Math Soc, 1982, 272(2): 753-769
[26]	Ramos M, Tavares H, Zou W. A Bahri-Lions theorem revisited. Adv Math, 2009, 222(6): 2173-2195
[27]	Rothschild L P, Stein E M. Hypoelliptic differential operators and nilpotent groups. Acta Math, 1976, 137(3/4): 247-320
[28]	Solimini S. Morse index estimates in min-max theorems. Manuscr Math, 1989, 63(4): 421-453
[29]	Struwe M. Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems. Manuscr Math, 1980, 32(3/4): 335-364
[30]	Struwe M. Variational Methods. New York: Springer, 2000
[31]	Tanaka K. Morse indices at critical points related to the symmetric mountain pass theorem and applications. Comm Partial Differ Equ, 1989, 14(1): 99-128
[32]	Yung P L. A sharp subelliptic Sobolev embedding theorem with weights. Bull Lond Math Soc, 2015, 47(3): 396-406