Sharp anisotropic singular Trudinger-Moser inequalities in the entire space

2023/05/21 by Kaiwen Guo, Yanjun Liu, Guo, Kaiwen +1 · 3 citations
Mathematics · #26D10 #35J70 #46E35 #Analytic and geometric function theory #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2305.12443

openalex publication_date 2023/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we investigate sharp singular Trudinger-Moser inequalities involving the anisotropic Dirichlet norm (∫_ΩF^N(∇ u) dx)^(1)/(N) in the Sobolev-type space D^N,q(ℝ^N), q≥ 1, here F:ℝ^N→[0,+∞) is a convex function of class C²(ℝ^N∖\0\), which is even and positively homogeneous of degree 1, its polar F⁰ represents a Finsler metric on ℝ^N. Combing with the connection between convex symmetrization and Schwarz symmetrization, we will establish anisotropic singular Trudinger-Moser inequalities and discuss their sharpness under several different situations, including the case ‖F(∇ u)‖_N≤ 1, the case ‖F(∇ u)‖_N^a+‖u‖_q^b≤ 1, and whether they are associated with exact growth.

Cited by

• Compactness of Extremals for Singular Anisotropic Trudinger-Moser functionals on bounded domain
• On the thresholds of the critical singular anisotropic Trudinger–Moser inequality
• Existence of extremal functions and Wulff symmetry for anisotropic Trudinger-Moser inequalities

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