$p-$Harmonic functions in the upper half-space

TL;DR

This paper studies p-harmonic functions in the upper half-space, focusing on their existence, nonexistence, symmetry, and boundary conditions for 1<p<N, using the method of moving planes.

Contribution

It provides new results on the existence, nonexistence, and symmetry of p-harmonic functions with nonlinear boundary conditions in the upper half-space.

Findings

Established conditions for existence and nonexistence of solutions.

Proved symmetry of positive solutions using the method of moving planes.

Analyzed qualitative properties of p-harmonic functions in the domain.

Abstract

This paper investigates the existence, nonexistence, and qualitative properties of p-harmonic functions in the upper half-space $\mathbb{R}^N_+ \, (N \geq 3)$ satisfying nonlinear boundary conditions for $1<p<N$. Moreover, the symmetry of positive solutions is shown by using the method of moving planes.