Let $\Omega\subset\mathbb R^n$, $u\in W^{1,p}(\Omega)$, $p>1$ satisfies $$\int_{\Omega}|\triangledown u|^{p-2}\triangledown u\triangledown\phi=0~~~,\forall\phi\in C_0^{\infty}(\Omega)$$ Then we call $u$ harmonic function.
When $p=n$, prove $u\in C^\alpha(\Omega)$
When $p<n$, prove $u\in C^{1,\alpha}(\Omega)$
I wonder how to prove two questions. The hints say that the former is proved by Caccioppoli inequality; the latter by Moser iteration.
Any advice is helpful. Thank you.