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Volume 37, Issue 4
A Weighted Trudinger-Moser Inequality and Its Extremal Functions in Dimension Two

Juan ZHAO & Pengxiu YU

DOI: 10.4208/jpde.v37.n4.3

J. Part. Diff. Eq., 37 (2024), pp. 402-416.

Published online: 2024-12

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  • Abstract

Let $Ω$ be a smooth bounded domian in $\mathbb{R}^2$ , $H^1_0 (Ω)$ be the standard Sobolev space, and $λ_f (Ω)$ be the first weighted eigenvalue of the Laplacian, namely, $$\lambda_f(\Omega)=\inf\limits_{u\in H^1_0(\Omega),\int_{\Omega}u^2{\rm dx}=1}\int_{\Omega}|\nabla u|^2f{\rm dx},$$where $f$ is a smooth positive function on $Ω.$ In this paper, using blow-up analysis, we prove$$\sup\limits_{u\in H^1_0(\Omega),\int_{\Omega}|\nabla u|^2f{\rm dx}\le 1}\int_{\Omega}e^{4\pi fu^2(1+\alpha||u||^2_2)}{\rm dx}<+\infty$$for any $0≤α<λ_f (Ω).$ Furthermore, extremal functions for the above inequality exist when $α>0$ is chosen sufficiently small.

  • Keywords

Trudinger-Moser inequality, extremal functions, blow-up analysis.

  • AMS Subject Headings

35J15, 46E35

  • Copyright

COPYRIGHT: © Global Science Press

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  • BibTex
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@Article{JPDE-37-402, author = {ZHAO , Juan and YU , Pengxiu}, title = {A Weighted Trudinger-Moser Inequality and Its Extremal Functions in Dimension Two}, journal = {Journal of Partial Differential Equations}, year = {2024}, volume = {37}, number = {4}, pages = {402--416}, abstract = {

Let $Ω$ be a smooth bounded domian in $\mathbb{R}^2$ , $H^1_0 (Ω)$ be the standard Sobolev space, and $λ_f (Ω)$ be the first weighted eigenvalue of the Laplacian, namely, $$\lambda_f(\Omega)=\inf\limits_{u\in H^1_0(\Omega),\int_{\Omega}u^2{\rm dx}=1}\int_{\Omega}|\nabla u|^2f{\rm dx},$$where $f$ is a smooth positive function on $Ω.$ In this paper, using blow-up analysis, we prove$$\sup\limits_{u\in H^1_0(\Omega),\int_{\Omega}|\nabla u|^2f{\rm dx}\le 1}\int_{\Omega}e^{4\pi fu^2(1+\alpha||u||^2_2)}{\rm dx}<+\infty$$for any $0≤α<λ_f (Ω).$ Furthermore, extremal functions for the above inequality exist when $α>0$ is chosen sufficiently small.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v37.n4.3}, url = {http://global-sci.org/intro/article_detail/jpde/23688.html} }
TY - JOUR T1 - A Weighted Trudinger-Moser Inequality and Its Extremal Functions in Dimension Two AU - ZHAO , Juan AU - YU , Pengxiu JO - Journal of Partial Differential Equations VL - 4 SP - 402 EP - 416 PY - 2024 DA - 2024/12 SN - 37 DO - http://doi.org/10.4208/jpde.v37.n4.3 UR - https://global-sci.org/intro/article_detail/jpde/23688.html KW - Trudinger-Moser inequality, extremal functions, blow-up analysis. AB -

Let $Ω$ be a smooth bounded domian in $\mathbb{R}^2$ , $H^1_0 (Ω)$ be the standard Sobolev space, and $λ_f (Ω)$ be the first weighted eigenvalue of the Laplacian, namely, $$\lambda_f(\Omega)=\inf\limits_{u\in H^1_0(\Omega),\int_{\Omega}u^2{\rm dx}=1}\int_{\Omega}|\nabla u|^2f{\rm dx},$$where $f$ is a smooth positive function on $Ω.$ In this paper, using blow-up analysis, we prove$$\sup\limits_{u\in H^1_0(\Omega),\int_{\Omega}|\nabla u|^2f{\rm dx}\le 1}\int_{\Omega}e^{4\pi fu^2(1+\alpha||u||^2_2)}{\rm dx}<+\infty$$for any $0≤α<λ_f (Ω).$ Furthermore, extremal functions for the above inequality exist when $α>0$ is chosen sufficiently small.

ZHAO , Juan and YU , Pengxiu. (2024). A Weighted Trudinger-Moser Inequality and Its Extremal Functions in Dimension Two. Journal of Partial Differential Equations. 37 (4). 402-416. doi:10.4208/jpde.v37.n4.3
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