Quantization property of n-Laplacian mean field equation and sharp Moser-Onofri inequality

Abstract

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 $\Omega$ is a smooth bounded domain of ${\mathbb{R}}^n(n\ge 2)$ and $-{\Delta }_{n}u=-\mathrm{div}(|\nabla u{|}^{n-2}\nabla u)$. 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 $B^n:=B^n(0,1)$, $$\underset{u\in W_0^{1,n}(B^n)}{\inf }\frac{1}{n{C}_n}{\int }_{B^n}|\nabla u{|}^{n}dx-\ln {\int }_{B^n}{e}^{u}dx\ge C(n),$$ which extends the result of Caglioti-Lions-Marchioro-Pulvirenti in \cite{Caglioti} to the case of $n$-dimensional ball. Here $C_n=(\frac{n^2}{n-1}{)}^{n-1}{\omega }_{n-1}$ and ${\omega }_{n-1}$ 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.

Some errors in the computational details of the test function in Part II of Section 3 have been revised, and the paper has been published in Mathematische Annalen

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