Polyharmonic function

Remarks

For $m = 1$ this is the definition of a harmonic function, and for $m = 2$ that of a solution of the biharmonic equation. A function that is polyharmonic of order $m$ is polyharmonic of every order $m' \ge m$. Example: $u(x) = \abs{x}^4$ on $\R^n$ is polyharmonic of order $3$: $\Delta \abs{x}^4 = 4(n+2)\abs{x}^2$, $\Delta^2 \abs{x}^4 = 8n(n+2)$ and $\Delta^3 \abs{x}^4 = 0$. Non-example: the same $u$ is not polyharmonic of order $2$, since $\Delta^2 u = 8n(n+2) \ne 0$. Non-example: $u(x) = e^{x_1}$ is polyharmonic of no order, since $\Delta^m u = e^{x_1} \ne 0$ for all $m$.