Nonlinear Neumann boundary problems for $n$-Laplacian Liouville equation on a half space

TL;DR

This paper classifies solutions to the nonlinear $n$-Laplacian Liouville equation with positive Neumann boundary conditions on a half-space, extending previous results from second order and subcritical cases to general $n$ and critical $p=n$.

Contribution

It generalizes the classification of solutions for the $n$-Laplacian Liouville equation with positive Neumann boundary conditions from lower dimensions and subcritical cases to all $n \\geq 2$ and the critical case $p=n$.

Findings

Extended classification results to all $n \\geq 2$ for the $n$-Laplacian Liouville equation.

Unified the understanding of solutions for critical $p$-Laplacian equations.

Provided a comprehensive solution classification under positive nonlinear Neumann boundary conditions.

Abstract

In this paper, for general $n\geq2$, we classify solutions to $n$-Laplacian Liouville equation with positive nonlinear Neumann boundary condition on the half-space $\mathbb{R}^{n}_{+}$. Under the positive nonlinear Neumann boundary condition, our result extend the classification result for the second order Liouville equation in \cite{Li} from $n=2$ to general $n\geq2$, and also extend the classification result for critical $p$-Laplacian equation in \cite{Zhou} from $p<n$ to $p=n$.