Prove or disprove the unique continuation of $p$-harmonic functions in $\mathbb{R}^n$, for $n\ge 3$.

Problem's Description

Consider a $p$-harmonic function in the unit ball $\mathbb{B}(0,1)\subset\mathbb{R}^n$, where $1< p< \infty$, $p\not=2$, and $n\ge3$. Suppose that $u(x)=0$ for all $x\in \mathbb{B}(0,1/2)$. Does it follow that $u$ is identically zero in $\mathbb{B}(0,1)$?

1. Article $p$-harmonic functions on the plane

   • Juan J Manfredi

   Proc. Amer. Math. Soc. 103 (2), 473--479, 1988