Lean on / lean against

Dears,

I always thought that one can say “to lean against a door” but not “to lean on a door”.
Could you please explain me why in some places they write the phrase “Do not lean on door” instead of “Do not lean against door”.
Which of those two variants is grammatically preferable?

And why there is no article before the word “door”?

TIA for your reply.

Do not lean on the door. / Do not lean against the door. are both in general use, but strictly speaking in formal language ‘against’ is correct.

There should be an article. If it is missing on a sign or similar, then it is because the writer has chosen to use a short, informal note rather than a grammatically correct sentence. This often happens with such notices because of a lack of space on the sign, or because a minimum of words makes the important part of the message stand out.