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

Acta mathematica scientia,Series A ›› 2026, Vol. 46 ›› Issue (2): 428-451.

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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)

Abstract

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

CLC Number:

O175.25

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.

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

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