Liouville Type Theorem For A Nonlinear Neumann Problem

TL;DR

This paper establishes a Liouville type theorem for a nonlinear Neumann problem involving a weighted divergence operator, using the moving plane method to derive symmetry and nonexistence results under certain conditions.

Contribution

It provides a new Liouville theorem for a class of nonlinear Neumann problems with weighted operators, extending previous results with novel applications.

Findings

Liouville theorem proved for the nonlinear Neumann problem

Symmetry results obtained via the moving plane method

Applications to nonexistence and classification of solutions

Abstract

Consider the following nonlinear Neumann problem \[ \begin{cases} \text{div}\left(y^{a}\nabla u(x,y)\right)=0, & \text{for }(x,y)\in\mathbb{R}_{+}^{n+1}\\ \lim_{y\rightarrow0+}y^{a}\frac{\partial u}{\partial y}=-f(u), & \text{on }\partial\mathbb{R}_{+}^{n+1},\\ u\ge0 & \text{in }\mathbb{R}_{+}^{n+1}, \end{cases} \] $a\in(-1,1)$. A Liouville type theorem and its applications are given under suitable conditions on $f$. Our tool is the famous moving plane method.