Are the derivatives of a $p$-harmonic function also a $p$ harmonic function? Since
\begin{eqnarray}
\Delta u_{x_j} &=& \sum_{i=1}^n (u_{x_j})_{x_i x_i}\\
&=& \sum_{i=1}^n (u_{x_j})_{x_ix_i}\\
&=& \left (\sum_{i=1}^n (u)_{x_i x_i}
\right )_{x_j}\\
&=& (\Delta u)_{x_j}=0.
\end{eqnarray}
So, the derivatives of harmonic functions are also harmonic functions.
Observe that harmonic fucntions is a $p$-harmonic fucntions when $p$ is equal to 2. However, the general case seens not to be so easy to calculate.
Remenber that a $p$-harmonic fucntion satisfyies
\begin{eqnarray}
\Delta_{p} u &: = &\texttt{div} \left ( | Du|^{p-2} Du) \right ) \\
& = & |Du|^{p-4} \left \{ | Du|^{2} \Delta u + (p-2) \sum_{i,j=1}^{n} u_{x_i} u_{x_j} u_{x_i x_j} \right \}.
\end{eqnarray}´
Based on calculations above is natural to ask if
\begin{eqnarray}
(\Delta_{p} u)_{x_j} =  (\Delta_{p} (u_{x_j})?)
\end{eqnarray}
 A: No. The fact that the $p$-Laplacian is nonlinear when $p\ne 2$ kills any hope for identities like $(\Delta_{p} u)_{x_j} =  \Delta_{p} (u_{x_j})$, and also makes the derivatives of $p$-harmonic functions unlikely to be $p$-harmonic, except for the simplest examples (affine functions).
Counterexample. It is known that $u(x)=|x|^{(p-n)/(p-1)}$ is $p$-harmonic in $\mathbb{R}^n\setminus\{0\}$. Let's take $p=4$ and $n=2$ here, so $u(x)=|x|^{2/3}= (x_1^2+x_2^2)^{1/3}$. The $p$-Laplacian simplifies to
$$\Delta_{4} u =   (u_{x_1}^2+u_{x_2}^2)(u_{x_1x_1} + u_{x_2x_2})  + 2 \sum_{i,j=1}^{2} u_{x_i} u_{x_j} u_{x_i x_j} \tag1 $$
Without relying on "It is known", one can check that for the above function $u$ the formula (1) yields
$$
\Delta_4 u = \frac{16}{81(x_1^2 + x_2^2)} - \frac{16}{81(x_1^2 + x_2^2)} = 0
$$
But for the function $$v = u_{x_1}=\frac{2 x_{1}}{3 \left(x_{1}^{2} + x_{2}^{2}\right)^{2/3 }}$$ we get
$$
\Delta_4 v = - \frac{64 x_{1} \left(x_{1}^{2} + 9 x_{2}^{2}\right)}{2187 \left(x_{1}^{2} + x_{2}^{2}\right)^{4}} +
\frac{64 x_{1}\left(x_{1}^2 - 9 x_{2}^2\right)}{2187 \left(x_{1}^{2} + x_{2}^{2}\right)^{4}} 
= - \frac{128 x_{1} x_{2}^{2}}{243 \left(x_{1}^{2} + x_{2}^{2}\right)^{4}}
$$
(Computation assisted by SymPy).
