= Solution
The local <Harnack inequality for harmonic functions> states that if $u\geq0$ is harmonic on a neighbourhood of $\overline{B_{4r}(a)}$, then
$$
\boxed{\sup_{B_r(a)}u\leq3^n\inf_{B_r(a)}u.}
$$
More generally, for any compact $K$ in a connected open domain $\Omega$, there is $C=C(K,\Omega,n)$ such that $\sup_Ku\leq C\inf_Ku$ for every nonnegative harmonic $u$ on $\Omega$. Positivity is essential; the assertion is not for arbitrary sign-changing functions.
First prove the <mean value property for harmonic functions>. For a smooth harmonic function the spherical mean has derivative
$$
m_x'(t)=\frac1{|\mathbb S^{n-1}|t^{n-1}}\int_{B_t(x)}\Delta u=0,
$$
so it equals $u(x)$; integration in the radius gives the ball mean property. The preceding <Weyl lemma> supplies smoothness if harmonicity was initially distributional.
For $x,z\in B_r(a)$ the inclusion $B_r(x)\subset B_{3r}(z)\subset B_{4r}(a)$ and nonnegativity give
$$
u(x)=\frac1{|B_r|}\int_{B_r(x)}u\leq\frac1{|B_r|}\int_{B_{3r}(z)}u=3^nu(z).
$$
Taking the supremum over $x$ and the infimum over $z$ proves the local estimate directly. To obtain the compact-set form, connect centres of a finite ball cover of $K$ to a fixed interior point by paths in $\Omega$. The compact union of those paths stays a positive distance from the complement of $\Omega$. Subdivide the paths into finitely many steps small enough that the local estimate applies at consecutive points. Multiply the finitely many comparison constants, first from one point of $K$ to the reference point and then to any other point of $K$. This gives a constant independent of $u$. In particular a nonnegative harmonic function is either everywhere positive or identically zero on each connected component.
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