1. Use gradient estimates for harmonic function,
|grad^k u(x)|<=max_{B(0, 2|x|)} |u(y)|/|x|^k
You can immediately show that grad^p u(x) is a bounded harmonic function, therefore a constant.
2. just by gradient estimates as well, assume R=1, then do the problem. Reference: Elliptic Partial Differential Equations by Qing Han&Fanghua Lin
|grad^k u(x)|<=max_{B(0, 2|x|)} |u(y)|/|x|^k
You can immediately show that grad^p u(x) is a bounded harmonic function, therefore a constant.
2. just by gradient estimates as well, assume R=1, then do the problem. Reference: Elliptic Partial Differential Equations by Qing Han&Fanghua Lin
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