Homogeneous polynomial $p$-harmonic functions

Problem's Description

Problem. Do there exist homogeneous degree $m\ge 2$ polynomial $p$-harmonic functions (i.e. solutions of $\,\,\mathrm{div} |\nabla u|^{p-2}\nabla u=0\,$), $x\in \mathbb{R}^n$ for some $p>1$ and $p\ne 2$?

The problem comes back to [1]. It is known that the answer is negative in the following cases:

$\bullet$ $n=2$ and any $m\ge 2$ [1];

$\bullet$ $m=3$ and any $n\ge 2$ [3];

$\bullet$ $m=4$ and any $n\ge 2$, $m=5$ and $n=3$ [2].

On the other hand, there exist rational homogeneous $p$-harmonic functions for certain pairs $(n,p)$, see [4].

1. Article Smoothness of certain degenerate elliptic equations

   • John L. Lewis

   Proceedings of the American Mathematical Society 80, 259-265, 1980fulltext

2. Article On p Laplace polynomial solutions

   • John L. Lewis,
   • Andrew Vogel

   The Journal of Analysis 24, 143-166, 2016fulltext

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

   • Vladimir G. Tkachev

   Proceedings of the American Mathematical Society 144, 2375-2382, 2016fulltext

4. Article New explicit solutions to the $p$-Laplace equation based on isoparametric foliations

   • V.G. Tkachev

   submitted, 2018arXiv