具有精确增长条件的各向异性Moser-Trudinger不等式的最佳常数及其应用

中文摘要 AI 辅助

我们研究了$\boldsymbol{\text{R}}^n$中Finsler度量框架下几类Moser-Trudinger不等式相关的最佳常数。通过采用无需凸对称化的方法，我们首先建立了临界与次临界各向异性奇异Moser-Trudinger不等式。进一步，我们在精确增长条件下研究了奇异与非奇异情形的各向异性Moser-Trudinger不等式，强调了最佳常数的最优性以及分母中出现的尖锐指数。为丰富该理论，我们引入了适配Finsler度量场景的拟共形型变换，以推导这类不等式的若干新变体。我们还刻画了精确增长条件下广义各向异性Moser-Trudinger不等式的极大元的可达性与不可达性，并明确确定了对应的上确界值。作为这些不等式的应用，我们分析了$\boldsymbol{\text{R}}^n$中一类含恒定势的$n$-Finsler-Laplace方程的正基态解的存在性与不存在性，该方程的非线性项在无穷远处呈现临界指数增长。

英文摘要

We investigate the best constants associated with several types of Moser--Trudinger inequalities within the framework of the Finsler metric in $\mathbb{R}^n$. By employing a convex-symmetrization-free approach, we first establish both the critical and subcritical anisotropic singular Moser--Trudinger inequalities. We further study anisotropic Moser--Trudinger inequalities in both singular and nonsingular cases under the exact growth condition, highlighting the optimality of the best constant as well as the sharp exponent appearing in the denominator. To further enrich the theory, we introduce quasi-conformal-type transformations adapted to the Finsler metric setting to derive several new variants of such inequalities. We also characterize the attainability and nonattainability of maximizers for a generalized version of anisotropic Moser--Trudinger inequalities with the exact growth and explicitly determine the corresponding supremum values. As an application of these inequalities, we analyze the existence and nonexistence of positive ground state solutions to a class of $n$-Finsler-Laplace equations involving constant potential in $\mathbb{R}^n$, where the nonlinearity exhibits a critical exponential growth at infinity.

发表机构

• Universidade de Brasília（巴西利亚大学）
• Indian Institute of Technology Kanpur（坎普尔印度理工学院）

机构由 AI 辅助整理，请以论文原文为准。

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75 pages