= Solution
Iteration gives the <interior derivative estimate for a harmonic function>
$$
|D^{k+1}u(x_0)|\leq C(n,k)R^{-(k+1)}\sup_{B_R(x_0)}|u|.
$$
The growth hypothesis bounds the right side by $C'R^{\alpha-1}+o(1)$, which tends to zero as $R\to\infty$. Thus every derivative of order $k+1$ vanishes everywhere. The <Taylor theorem> makes $u$ a polynomial of degree at most $k$, proving the <Polynomial-growth Liouville theorem for harmonic functions>.
Solved by gpt-5.6-sol high.
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