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

TL;DR

This work establishes a quantization phenomenon for the $n$-Laplacian mean field equation $-\Delta_n u=\lambda e^u$ and links it to a sharp Moser-Onofri constant in the unit ball. It proves blow-up occurs at finitely many points with per-point mass $ (\frac{n}{n-1}\alpha_n)^{n-1}$ and derives the limiting equation $-\Delta_n u_0=\sum_i (\frac{n}{n-1}\alpha_n)^{n-1}\delta_{x_i}$, while identifying the blow-up mass via Pohozaev-type identities. For the ball, the paper shows nonexistence of extremals and computes the exact infimum using subcritical approximation and capacity estimates, with the asymptotics captured by a Radon–Nikodym limit described by $\eta_0$. Extending to general domains, it develops an optimal concentration framework through $n$-harmonic transplantation and defines a criterion involving the $n$-harmonic radius and Robin function to determine existence of extremals, thereby bridging local concentration and global geometry. Overall, the results unify blow-up analysis, sharp constants, and geometric-analytic criteria for extremals in higher-dimensional Moser-Onofri-type inequalities.

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 $Ω$ 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.

Theorems & Definitions (33)

• Theorem 1.1
• Remark 1.2
• Corollary 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 1.8
• Lemma 2.1
• Lemma 2.2