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

By using a nonassociative algebra argument, we prove that $u\equiv 0$ is the only cubic homogeneous polynomial solution to the $p$-Laplace equation $\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{\mid }Du{\mathrm{\mid }}^{p-2}Du(x)=0$ in ${\mathbb{R}}^n$ for any $n\ge 2$ and $p\in ̸\{0,2\}$.