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Extremals for a Hardy–Trudinger–Moser Inequality with Remainder Terms

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Abstract

Let \(B\subset {\mathbb {R}}^2\) be the unit disk , \({\mathcal {H}}\) be the completion of \(C_{0}^{\infty }(B)\) under the norm

$$\begin{aligned} ||u||_{{\mathcal {H}}}=\bigg (\int _{B}|\nabla u|^2\textrm{d}x-\int _{B}\frac{u^2}{(1-|x|^2)^2}\textrm{d}x \bigg )^{1/2}. \end{aligned}$$

In this paper, we consider a maximum problem concerning the Hardy–Trudinger–Moser inequalities containing lower order perturbation. Namely, there exists a positive constant \(\varepsilon _{0}\) such that if \(\gamma \le 4\pi +\varepsilon _{0}\), then

$$\begin{aligned} \sup \limits _{u\in {\mathcal {H}},\;|| u||_{{\mathcal {H}}}\le 1}\int _{B}(e^{4\pi u^{2}}-\gamma u^2)\textrm{d}x \end{aligned}$$

can be achieved by some functions \(u_{0}\in {\mathcal {H}}\) with \(||u_{0}||_{{\mathcal {H}}}=1\). This expands the results of Wang and Ye (Adv Math 230:294–320, 2012).

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Authors and Affiliations

  1. School of Mathematics, Renmin University of China, Beijing, 100872, China

    Kexin Yin

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Communicated by Saeid Maghsoudi.

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Yin, K. Extremals for a Hardy–Trudinger–Moser Inequality with Remainder Terms. Bull. Iran. Math. Soc. 49, 64 (2023). https://doi.org/10.1007/s41980-023-00813-4

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  • DOI: https://doi.org/10.1007/s41980-023-00813-4

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