Extremal Functions for a Trudinger-Moser Inequality with a Sign-Changing Weight

IF 1 3区数学 Q1 MATHEMATICS

Potential Analysis Pub Date : 2024-07-20 DOI:10.1007/s11118-024-10159-z

Pengxiu Yu, Xiaobao Zhu

引用次数: 0

Abstract

Let $(\Sigma ,g)$ be a closed Riemann surface, $\lambda _1(\Sigma )$ be the first eigenvalue of the Laplace-Beltrami operator. Assume $h:\Sigma \rightarrow \mathbb {R}$ is some smooth sign-changing function. Using blow-up analysis, we prove that for any $\alpha <\lambda _1(\Sigma )$, the supremum

$$\sup _{\int _\Sigma |\nabla _gu|^2dv_g-\alpha \int _\Sigma u^2dv_g\le 1,\,\int _\Sigma udv_g=0}\int _\Sigma he^{4\pi u^2}dv_g$$

is attained by some admissible function $u_\alpha $. This generalizes earlier results of Yang (J. Differential Equations 2015) and Hou (J. Math. ineq. 2018). Our result resembles existence of solutions to the mean field equations

$$\Delta _gu=8\pi \left( \frac{he^u}{\int _\Sigma he^udv_g}-\frac{1}{|\Sigma |}\right) ,$$

where h is a smooth sign-changing function. Such problems were extensively studied by L. Sun and J. Y. Zhu (Cal. Var. 2021; arXiv: 2012.12840).

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带有符号变化权重的特鲁丁格-莫泽尔不等式的极值函数

让（(\Sigma ,g)\) 是一个封闭的黎曼曲面，$\lambda _1(\Sigma )$ 是拉普拉斯-贝尔特拉米算子的第一个特征值。假设（h:\Sigma \rightarrow \mathbb {R}）是某个平滑的符号变化函数。通过吹胀分析，我们可以证明对于任何 $α <；\1(\Sigma )$, the supremum $$\sup _{int _\Sigma |\nabla _gu|^2dv_g\alpha \int _\Sigma u^2dv_g\le 1、\,\int _\Sigma udv_g=0}\int _\Sigma he^{4\pi u^2}dv_g$$ 是通过某个可接受的函数 $u_\alpha $ 达到的。这概括了 Yang (J. Differential Equations 2015) 和 Hou (J. Math. ineq. 2018) 的早期结果。我们的结果类似于均值场方程的解的存在性 $$\Delta _gu=8\pi \left( \frac{he^u}{\int _\Sigma he^udv_g}-\frac{1}{|\Sigma |}\right) ,$$where h is a smooth sign changing function.L. Sun 和 J. Y. Zhu 对此类问题进行了广泛研究 (Cal. Var. 2021; arXiv: 2012.12840)。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Potential Analysis 数学-数学

CiteScore

2.20

自引率

9.10%

发文量

审稿时长

>12 weeks

期刊介绍： The journal publishes original papers dealing with potential theory and its applications, probability theory, geometry and functional analysis and in particular estimations of the solutions of elliptic and parabolic equations; analysis of semi-groups, resolvent kernels, harmonic spaces and Dirichlet forms; Markov processes, Markov kernels, stochastic differential equations, diffusion processes and Levy processes; analysis of diffusions, heat kernels and resolvent kernels on fractals; infinite dimensional analysis, Gaussian analysis, analysis of infinite particle systems, of interacting particle systems, of Gibbs measures, of path and loop spaces; connections with global geometry, linear and non-linear analysis on Riemannian manifolds, Lie groups, graphs, and other geometric structures; non-linear or semilinear generalizations of elliptic or parabolic equations and operators; harmonic analysis, ergodic theory, dynamical systems; boundary value problems, Martin boundaries, Poisson boundaries, etc.