Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 107 / 1 / c

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 $v(x)\le 3^{n}v(x_0)$ for every $x$. Thus $v$ is globally bounded. The interior derivative estimate for a harmonic function now gives $|Dv(x)|\le C\Vert v{\Vert }_{\infty }/R$ 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.