= Solution
Work first with essential bounds. On a <Euclidean ball> $B_R(x_0)\Subset B_1$, put $M=\operatorname*{ess\,sup}_{B_R}u$, $m=\operatorname*{ess\,inf}_{B_R}u$ and $\omega=M-m$. The functions $M-u$ and $u-m$ are nonnegative <weak solutions>, hence <weak supersolutions>. In $B_{R/2}$, at least one of the sets $\{u\leq(M+m)/2\}$ and $\{u\geq(M+m)/2\}$ has at least half the measure.
In the first case, the <Weak Harnack inequality> applied to $M-u$ gives
$$
C_H\bigl(M-\operatorname*{ess\,sup}_{B_{R/2}}u\bigr)
\geq\left(\frac1{|B_{R/2}|}\int_{B_{R/2}}(M-u)^q\right)^{1/q}
\geq2^{-1-1/q}\omega.
$$
In the second case, applying it to $u-m$ gives the same lower bound for the improvement of the minimum. Thus the <oscillation decay estimate> is
$$
\boxed{\operatorname{osc}_{B_{R/2}}u\leq\tau\operatorname{osc}_{B_R}u,
\qquad \tau=1-C_H^{-1}2^{-1-1/q}\in(0,1).}
$$
This measure argument works even when $q<1$, avoiding a false use of the triangle inequality in that range.
Iteration yields $\operatorname{osc}_{B_r(x_0)}u\leq C(r/R)^\mu\operatorname{osc}_{B_R(x_0)}u$, with, for example, $\mu=\min\{1/2,-\log\tau/\log2\}\in(0,1)$. At <Lebesgue points> the shrinking oscillations define a unique continuous representative, and the estimate extends to every point by continuity. For $x,y\in B_{1/4}$ at distance less than $1/4$, apply the estimate to a <Euclidean ball> centred at $x$ with initial radius $1/2$ and final radius comparable to $|x-y|$. This <Euclidean ball> stays in $B_1$. Larger distances use the trivial bound $2\|u\|_\infty$. We obtain
$$
\boxed{[u]_{C^{0,\mu}(\overline{B_{1/4}})}\leq C\|u\|_{L^\infty(B_1)},
\qquad u\in C^{0,\mu}(\overline{B_{1/4}}).}
$$
The constants depend only on $n,\lambda,\Lambda$. This derives the relevant conclusion of the <De Giorgi-Nash-Moser theorem> directly from part (a).
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