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Volume 36, Issue 4
Extremal Functions for an Improved Trudinger-Moser Inequality Involving $L^p$-Norm in $\mathbb{R}^n$

Liu Yang & Xiaomeng Li

DOI: 10.4208/jpde.v36.n4.7

J. Part. Diff. Eq., 36 (2023), pp. 414-434.

Published online: 2023-11

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  • Abstract

Let $W^{1,n} (\mathbb{R}^n)$ be the standard Sobolev space. For any $\tau>0$ and $p>n>2,$ we denote $$\lambda_{n,p}=\inf\limits_{u\in W^{1,n}(\mathbb{R}^n),\ u\not\equiv 0}\frac{\int_{\mathbb{R}^n}(|∇u|^n+\tau|u|^n)dx}{(\int_{\mathbb{R}^n}|u|^p dx)^{\frac{n}{p}}}. $$ Define a norm in $W^{1,n} (\mathbb{R}^n)$ by $$||u||_{n,p}=(\int_{\mathbb{R}^n}(|∇u|^n+\tau|u|^n)dx-\alpha(\int_{\mathbb{R}^n}|u|^pdx)^{\frac{n}{p}})^{\frac{1}{n}},$$ where $0 ≤ α < λ_{n,p}.$ Using a rearrangement argument and blow-up analysis, we will prove $$\sup\limits_{u\in W^{1,n}(\mathbb{R}^n), \ ||u||_{n,p}\le 1}\int_{\mathbb{R}^n}\left(e^{\alpha_n |u|^{\frac{n}{n-1}}}-\sum^{n-1}_{j=0}\frac{\alpha_n^j|u|^{\frac{jn}{n-1}}}{j!} \right)dx$$ can be attained by some function $u_0 ∈ W^{1,n} (\mathbb{R}^n )∩C^1 (\mathbb{R}^n)$ with $∥u_0∥_{n,p}= 1,$ here $α_n = nω ^{\frac{1}{ n−1}}_{n−1}$ and $ω_{n−1}$ is the measure of the unit sphere in $\mathbb{R}^n .$

  • Keywords

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

  • AMS Subject Headings

46E35

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-36-414, author = {Yang , Liu and Li , Xiaomeng}, title = {Extremal Functions for an Improved Trudinger-Moser Inequality Involving $L^p$-Norm in $\mathbb{R}^n$}, journal = {Journal of Partial Differential Equations}, year = {2023}, volume = {36}, number = {4}, pages = {414--434}, abstract = {

Let $W^{1,n} (\mathbb{R}^n)$ be the standard Sobolev space. For any $\tau>0$ and $p>n>2,$ we denote $$\lambda_{n,p}=\inf\limits_{u\in W^{1,n}(\mathbb{R}^n),\ u\not\equiv 0}\frac{\int_{\mathbb{R}^n}(|∇u|^n+\tau|u|^n)dx}{(\int_{\mathbb{R}^n}|u|^p dx)^{\frac{n}{p}}}. $$ Define a norm in $W^{1,n} (\mathbb{R}^n)$ by $$||u||_{n,p}=(\int_{\mathbb{R}^n}(|∇u|^n+\tau|u|^n)dx-\alpha(\int_{\mathbb{R}^n}|u|^pdx)^{\frac{n}{p}})^{\frac{1}{n}},$$ where $0 ≤ α < λ_{n,p}.$ Using a rearrangement argument and blow-up analysis, we will prove $$\sup\limits_{u\in W^{1,n}(\mathbb{R}^n), \ ||u||_{n,p}\le 1}\int_{\mathbb{R}^n}\left(e^{\alpha_n |u|^{\frac{n}{n-1}}}-\sum^{n-1}_{j=0}\frac{\alpha_n^j|u|^{\frac{jn}{n-1}}}{j!} \right)dx$$ can be attained by some function $u_0 ∈ W^{1,n} (\mathbb{R}^n )∩C^1 (\mathbb{R}^n)$ with $∥u_0∥_{n,p}= 1,$ here $α_n = nω ^{\frac{1}{ n−1}}_{n−1}$ and $ω_{n−1}$ is the measure of the unit sphere in $\mathbb{R}^n .$

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v36.n4.7}, url = {http://global-sci.org/intro/article_detail/jpde/22138.html} }
TY - JOUR T1 - Extremal Functions for an Improved Trudinger-Moser Inequality Involving $L^p$-Norm in $\mathbb{R}^n$ AU - Yang , Liu AU - Li , Xiaomeng JO - Journal of Partial Differential Equations VL - 4 SP - 414 EP - 434 PY - 2023 DA - 2023/11 SN - 36 DO - http://doi.org/10.4208/jpde.v36.n4.7 UR - https://global-sci.org/intro/article_detail/jpde/22138.html KW - Trudinger-Moser inequality, extremal function, blow-up analysis. AB -

Let $W^{1,n} (\mathbb{R}^n)$ be the standard Sobolev space. For any $\tau>0$ and $p>n>2,$ we denote $$\lambda_{n,p}=\inf\limits_{u\in W^{1,n}(\mathbb{R}^n),\ u\not\equiv 0}\frac{\int_{\mathbb{R}^n}(|∇u|^n+\tau|u|^n)dx}{(\int_{\mathbb{R}^n}|u|^p dx)^{\frac{n}{p}}}. $$ Define a norm in $W^{1,n} (\mathbb{R}^n)$ by $$||u||_{n,p}=(\int_{\mathbb{R}^n}(|∇u|^n+\tau|u|^n)dx-\alpha(\int_{\mathbb{R}^n}|u|^pdx)^{\frac{n}{p}})^{\frac{1}{n}},$$ where $0 ≤ α < λ_{n,p}.$ Using a rearrangement argument and blow-up analysis, we will prove $$\sup\limits_{u\in W^{1,n}(\mathbb{R}^n), \ ||u||_{n,p}\le 1}\int_{\mathbb{R}^n}\left(e^{\alpha_n |u|^{\frac{n}{n-1}}}-\sum^{n-1}_{j=0}\frac{\alpha_n^j|u|^{\frac{jn}{n-1}}}{j!} \right)dx$$ can be attained by some function $u_0 ∈ W^{1,n} (\mathbb{R}^n )∩C^1 (\mathbb{R}^n)$ with $∥u_0∥_{n,p}= 1,$ here $α_n = nω ^{\frac{1}{ n−1}}_{n−1}$ and $ω_{n−1}$ is the measure of the unit sphere in $\mathbb{R}^n .$

Yang , Liu and Li , Xiaomeng. (2023). Extremal Functions for an Improved Trudinger-Moser Inequality Involving $L^p$-Norm in $\mathbb{R}^n$. Journal of Partial Differential Equations. 36 (4). 414-434. doi:10.4208/jpde.v36.n4.7
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