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

Wei Dai, Changfeng Gui, Yichen Hu, Shaolong Peng

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 $p=n$.

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

Abstract

In this paper, for general , we classify solutions to -Laplacian Liouville equation with positive nonlinear Neumann boundary condition on the half-space . Under the positive nonlinear Neumann boundary condition, our result extend the classification result for the second order Liouville equation in \cite{Li} from to general , and also extend the classification result for critical -Laplacian equation in \cite{Zhou} from to .
Paper Structure (12 sections, 20 theorems, 266 equations)

This paper contains 12 sections, 20 theorems, 266 equations.

Table of Contents

  1. Introduction
  2. Background and setting of the problem
  3. Main results
  4. Crucial ingredients and sketch of our proof
  5. Structure of the paper
  6. Some notation
  7. Preliminaries
  8. Behavior of $u$ and $\nabla u$ at infinity
  9. Second-order regularity for weak solutions
  10. Proof of our main Theorem \ref{['Th:1-1']}
  11. The existence of the solution of $n$-Laplacian equation with mixed boundary condition
  12. Proof of Lemma \ref{['rele:2']}

Key Result

Theorem 1.1

Assume $n\geq2$. Any solution $u$ to equation eq:1-2 must be of the form for some $x^0\in \mathbb{R}^{n}_{-}$ and $\lambda=-\frac{c_{n}^{\frac{1}{n}}}{nx^{0}_{n}}>0$, where $c_{n}=n(\frac{n^2}{n-1})^{n-1}$.

Theorems & Definitions (37)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2: Brezis-Merle type exponential inequality with Neumann boundary condition
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4: Gobal $L^{\infty}$-estimate for mixed boundary condition
  • proof
  • ...and 27 more