Strong comparison Principle for $p$-harmonic functions

Problem's Description

Suppose that $1< p< \infty$, $p \not= 2$ , and that the space dimension $n\ge 3$. Let $u$ and $v$ be $p$-harmonic functions in the unit ball $\mathbb{B}^n$ of $\mathbb{R}^n$. Suppose that for all $x\in\mathbb{B}^n$ we have $u(x)\le v(x)$ and that $u(0)=v(0)$. Prove that $u\equiv v$.

1. Article $p$-Harmonic Functions in the Plane

   • Juan J. Manfrdi

   Proceedings of the American Mathematical Society 103 (2), 473--479, 1988