= Solution
Shift $u$ by a constant and, if necessary, change sign to obtain an entire nonnegative <harmonic function> $v$. The <Harnack inequality for harmonic functions> on arbitrarily large balls centred at $x_0$ gives $v(x)\leq3^nv(x_0)$ for every $x$. Thus $v$ is globally bounded. The <interior derivative estimate for a harmonic function> now gives $|Dv(x)|\leq C\|v\|_\infty/R$ on balls of arbitrarily large radius $R$. Letting $R\to\infty$ proves $Dv=0$, hence $u$ is constant. This proves the one-sided <harmonic Liouville theorem> in every dimension.
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