Heisenberg 群上 Kohn-Laplace 方程解的存在性——献给陈化教授 70 寿辰

数学物理学报 ›› 2026, Vol. 46 ›› Issue (2): 669-682.

Heisenberg 群上 Kohn-Laplace 方程解的存在性——献给陈化教授 70 寿辰

张明()

浙江科技大学理学院杭州 310023

收稿日期:2025-12-30 修回日期:2026-03-10 出版日期:2026-04-26 发布日期:2026-04-27
作者简介:张明, Email：mingzhang_math@zust.edu.cn
基金资助:
国家自然科学基金(12571249)

Existence of Solution to the Kohn-Laplace Equation on the Heisenberg Group

Ming Zhang()

Faculty of Science, Zhejiang University of Science and Technology, Hangzhou 310023

Received:2025-12-30 Revised:2026-03-10 Online:2026-04-26 Published:2026-04-27
Supported by:
NSFC(12571249)

摘要/Abstract

摘要：

该文利用变分方法研究了 Heisenberg 群上一类含有临界 Sobolve 增长与对数非线性项的退化型偏微分方程正解的存在性. 与 [Brézis H, Nirenberg L. Comm Pure Appl Math, 1983, 36(4): 437-477] 问题中经典的扰动项 $\lambda u$ 相比, 对数非线性项 $\lambda u\log u^2$ 的出现缓和了临界增长引起的紧性障碍, 拓宽了正解存在的参数区间.

关键词: Heisenberg 群, 临界 Sobolev 指数, 对数非线性项, 变分方法

Abstract:

In this paper, we investigate, via variational methods, the existence of positive solution to a class of degenerate partial differential equations on the Heisenberg group involving critical Sobolev growth and logarithmic nonlinearity. In contrast to the classical perturbation term $\lambda u$ appearing in the [Brézis H, Nirenberg L. Comm Pure Appl Math, 1983, 36(4): 437-477], the logarithmic nonlinearity $\lambda u \log u^2$ leads to a weaker constraint on the parameter, yielding an improved existence result in the critical case.

Key words: Heisenberg group, critical Sobolev exponent, logarithmic nonlinearity, variational methods

中图分类号:

O175

张明. Heisenberg 群上 Kohn-Laplace 方程解的存在性——献给陈化教授 70 寿辰[J]. 数学物理学报, 2026, 46(2): 669-682.

Ming Zhang. Existence of Solution to the Kohn-Laplace Equation on the Heisenberg Group[J]. Acta mathematica scientia,Series A, 2026, 46(2): 669-682.

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