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Sharp anisotropic singular Trudinger–Moser inequalities in the entire space

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Abstract

In this paper, we investigate sharp singular Trudinger–Moser inequalities involving the anisotropic Dirichlet norm \(\left( \int _{{\mathbb {R}}^{N}}F^{N}(\nabla u)\;\textrm{d}x\right) ^{\frac{1}{N}}\) in Sobolev-type space \(D^{N,q}(\mathbb {R}^{N})\), \(N\ge 2\), \(q\ge 1\). Here \(F:\mathbb {R}^{N}\rightarrow [0,+\infty )\) is a convex function of class \(C^{2}(\mathbb {R}^{N}\setminus \{0\})\), which is even and positively homogeneous of degree 1. Combing with the connection between convex symmetrization and Schwarz symmetrization, we will establish anisotropic singular Trudinger–Moser inequalities and discuss their sharpness under different situations, including the case \(\Vert F(\nabla u)\Vert _{N}\le 1\), the case \(\Vert F(\nabla u)\Vert _{N}^{a}+\Vert u\Vert _{q}^{b}\le 1\), and whether they are associated with exact growth.

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Authors and Affiliations

  1. Chern Institute of Mathematics, Nankai University, Tianjin, 300071, People’s Republic of China

    Kaiwen Guo

  2. School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, People’s Republic of China

    Yanjun Liu

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  1. Kaiwen Guo
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  2. Yanjun Liu
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Correspondence to Yanjun Liu.

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Communicated by A. Mondino

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This work is supported by the National Natural Science Foundation of China (12201089, 12371051), the Natural Science Foundation Project of Chongqing (CSTB2022NSCQ-MSX0226), the Science and Technology Research Program of Chongqing Municipal Education Commission (KJQN202200513), the Innovation projects for studying abroad and returning to China(cx2023097) and Chongqing Normal University Foundation.

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Guo, K., Liu, Y. Sharp anisotropic singular Trudinger–Moser inequalities in the entire space. Calc. Var. 63, 82 (2024). https://doi.org/10.1007/s00526-024-02700-0

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