Quantization of blow-up masses for the Finsler $N$-Liouville equation

TL;DR

This paper establishes quantization of blow-up masses for the Finsler N-Liouville equation, extending classical and recent results to a more general anisotropic setting in higher dimensions.

Contribution

It generalizes existing blow-up mass quantization results to the Finsler N-Liouville equation, encompassing anisotropic and higher-dimensional cases.

Findings

Quantization of blow-up masses for the Finsler N-Liouville equation.

Extension of classical Liouville results to anisotropic Finsler settings.

Connections to previous work on Liouville and N-Laplacian equations.

Abstract

The quantization results for blow-up phenomena play crucial roles in the analysis of partial differential equations. Here we quantify the blow-up masses to the following Finsler $N$-Liouville equation $$-Q_{N}u_{n}=V_{n}e^{u_{n}}\quad\mbox{in}~ \Omega\subset \mathbb{R}^{N}, N \ge 2.$$ Our study generalizes the classical result of Li-Shafrir [Indiana Univ. Math.J.,1994] for Liouville equation, Wang-Xia's work for anisotropic Liouville equation in $\mathbb{R}^2$ [JDE, 2012], and Esposito-Lucia's for the $N$-Laplacian case in $\mathbb{R}^N$ ($N \geq 3$) in their recent paper [CVPDE, 2024].