Quantization property of n-Laplacian mean field equation and sharp Moser-Onofri inequality
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arXiv
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| Natura: | Preprint |
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2024
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| author | Chen, Lu Lu, Guozhen Wang, Bohan |
| author_facet | Chen, Lu Lu, Guozhen Wang, Bohan |
| contents | In this paper, we are concerned with the following $n$-Laplacian mean field equation \[ \left\{ {\begin{array}{*{20}{c}} { - Δ_n u = λe^u} & {\rm in} \ \ Ω, \\ {\ \ \ \ u = 0} &\ {\rm on}\ \partial Ω, \end{array}} \right. \] \[\] where $Ω$ is a smooth bounded domain of $\mathbb{R}^n \ (n\geq 2)$ and $- Δ_n u =-{\rm div}(|\nabla u|^{n-2}\nabla u)$. We first establish the quantization property of solutions to the above $n$-Laplacian mean field equation. As an application, combining the Pohozaev identity and the capacity estimate, we obtain the sharp constant $C(n)$ of the Moser-Onofri inequality in the $n$-dimensional unit ball $B^n:=B^n(0,1)$, $$\mathop {\inf }\limits_{u \in W_0^{1,n}(B^n)}\frac{1}{ n C_n}\int_{B^n} | \nabla u|^n dx- \ln \int_{B^n} {e^u} dx\geq C(n),$$ which extends the result of Caglioti-Lions-Marchioro-Pulvirenti in \cite{Caglioti} to the case of $n$-dimensional ball. Here $C_n=(\frac{n^2}{n-1})^{n-1} ω_{n-1}$ and $ω_{n-1}$ is the surface measure of $B^n$. For the Moser-Onofri inequality in a general bounded domain of $\mathbb{R}^n$, we apply the technique of $n$-harmonic transplantation to give the optimal concentration level of the Moser-Onofri inequality and obtain the criterion for the existence and non-existence of extremals for the Moser-Onofri inequality. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_00743 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantization property of n-Laplacian mean field equation and sharp Moser-Onofri inequality Chen, Lu Lu, Guozhen Wang, Bohan Analysis of PDEs In this paper, we are concerned with the following $n$-Laplacian mean field equation \[ \left\{ {\begin{array}{*{20}{c}} { - Δ_n u = λe^u} & {\rm in} \ \ Ω, \\ {\ \ \ \ u = 0} &\ {\rm on}\ \partial Ω, \end{array}} \right. \] \[\] where $Ω$ is a smooth bounded domain of $\mathbb{R}^n \ (n\geq 2)$ and $- Δ_n u =-{\rm div}(|\nabla u|^{n-2}\nabla u)$. We first establish the quantization property of solutions to the above $n$-Laplacian mean field equation. As an application, combining the Pohozaev identity and the capacity estimate, we obtain the sharp constant $C(n)$ of the Moser-Onofri inequality in the $n$-dimensional unit ball $B^n:=B^n(0,1)$, $$\mathop {\inf }\limits_{u \in W_0^{1,n}(B^n)}\frac{1}{ n C_n}\int_{B^n} | \nabla u|^n dx- \ln \int_{B^n} {e^u} dx\geq C(n),$$ which extends the result of Caglioti-Lions-Marchioro-Pulvirenti in \cite{Caglioti} to the case of $n$-dimensional ball. Here $C_n=(\frac{n^2}{n-1})^{n-1} ω_{n-1}$ and $ω_{n-1}$ is the surface measure of $B^n$. For the Moser-Onofri inequality in a general bounded domain of $\mathbb{R}^n$, we apply the technique of $n$-harmonic transplantation to give the optimal concentration level of the Moser-Onofri inequality and obtain the criterion for the existence and non-existence of extremals for the Moser-Onofri inequality. |
| title | Quantization property of n-Laplacian mean field equation and sharp Moser-Onofri inequality |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2406.00743 |