I am going to summarize some known comparison principles. I will start with a weak comparison principle for -Laplacian with Neumann boundary condition. This is adapted from a recent paper by J. Giacomoni et al. published in Differential Integral Equations [here]. I will keep the numbering for the convenience.
Lemma 3.1. Let be non-negative functions satisfying
on the boundary in the strong sense. Then in .
Proof. The trick is that if we want to prove in then we just show that
in . To this purpose, we use as a test function in the equation
and integrate to achive
and use the integration by parts to arrive at
Keep in mind that
We now use the following inequality
that proof can be found in this topic. To be precise, we get
This proves the positivity of the first term. For the second term, it is clear to see that
Lastly, we know that the integral on is non-positive since , thus the left hand side is non-negative. This gives the fact that .