Let $B=B_1(0)\subset \mathbb R^N$ and let $u\geq 0$ solve the PDE $$ -\Delta_p u=1\,\,\mbox{in $B$} $$
Let another function $f$ be such that $$ \begin{cases} -\Delta_p f =1 \;\;\mbox{in $B$}\\ f=0 \;\;\mbox{on $\partial B$} \end{cases} $$
I wish to show that in $B_{1/2}=B_{1/2}(0)$, $u-f$ satify a Harnack type estimate : $$ \frac{1}{C}(u-f)(0)\leq (u-f)\leq C (u-f)(0) $$ For some $C$, an scaling invariant constant depending only on the dimension.
PS: it shall lead to same kind of estimates on $u$ because $f$ can be found explicitly as $f(x)=c(1-|x|^{p/(p-1)})$, $c$ is adjusted so that $f$ satisfy the given PDE.
Harnack holds for $p$-harmonic functions ($\Delta_p u = 0$), but here $u-f$ is not $p$-harmonic since $\Delta_p$ is not linear. What you ask is related (maybe implies?) a strong comparison principle for the $p$-Laplace equation, which is an open problem for $N\geq 3$. A good reference for this is the notes on the $p$-Laplacian by Peter Lindqvist, which you can find on google.