Are the derivatives of a $p$-harmonic function also a $p$ harmonic function?

Question

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}

Answer 1

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).