"I have a small apartment."
Translation:У мене маленька квартира.
Since it's "I have", in principle the "є" is there.
Ordering by the strength of ownership:
"Я маю": in central Ukraine or in "textbook" Ukrainian, sounds really strong, almost like "I own this apartment". In the West is a normal way to say a neutral "I have".
"У мене є": Not so strong as "я маю", more natural in central Ukraine. Simply means you have an apartment.
"У мене": if you don't skip "є" it sounds like you have bought the apartment. Skipping it makes it more ambiguous, could be that you are just renting it (most likely, otherwise you wouldn't have skipped "є"...)