Let $\Omega\subset \mathbb{R}^n$ be a smooth domain and $u\in W^{1,p}(\Omega)$ a non-constant $p$-harmonic function, for some $1<p<n$.
 
**Question:** What is the Hausdorff dimension of the critical set
$$
\Sigma(u)=\{x\in \Omega\mid \nabla u(x)=0\}?
$$
For example, can the dimension be bounded above by $n-p$?

(Note that in general the optimal regularity of $u$ is $C^{1,\alpha}_{\mathrm{loc}}(\Omega)$, and if $p$ satisfies the Cordes condition $1<p<3+\frac{2}{n-2}$, then $u$ is also $W^{2,2}_{\mathrm{loc}}(\Omega)$. See [Lindqvist's notes][1].)

**Some results I have found:** (In a [review article][2] by Magnanini)

 - When $n=2$ and $1<p<\infty$ (the planar case), it is known that $\Sigma(u)$ consists of isolated points. So this case is solved. The proofs rely on complex function methods, which I don't see generalizing to higher dimensions.
 - When $n\geq 3$ and $p=2$ (the linear case), it is known that $\dim_H \Sigma(u)\leq n-2$. The proofs I've seen for this case rely on the Bers approximation (locally approximating a harmonic function by harmonic polynomials), which I am not sure how to generalize to other values of $p$.

References would be appreciated.

  [1]: https://jyx.jyu.fi/bitstream/handle/123456789/54658/1/Lindqvistplaplacenotes.pdf
  [2]: https://arxiv.org/pdf/1604.00530.pdf