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

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research 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

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

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