On the non-vanishing property for real analytic solutions of the $p$-Laplace equation

TL;DR

This paper proves that the only cubic homogeneous polynomial solution to the p-Laplace equation in any dimension, for p not equal to 0 or 2, is the trivial solution, using a nonassociative algebra approach.

Contribution

It introduces a novel algebraic method to analyze polynomial solutions of the p-Laplace equation, establishing a non-vanishing property for cubic solutions.

Findings

Only the trivial solution exists for cubic homogeneous polynomials.

The method applies to all dimensions n ≥ 2.

Valid for p ≠ 0, 2.

Abstract

By using a nonassociative algebra argument, we prove that $u\equiv0$ is the only cubic homogeneous polynomial solution to the $p$-Laplace equation $\mathrm{div} |Du|^{p-2}Du(x)=0 $ in $\mathbb{R}^n$ for any $n\ge2$ and $p\not\in\{0,2\}$.