This may be more of an opinion than fact, but I can't leave comments so..
Existence of X is a relation between X and either a specifically mentioned set S, or if left out, implicitly usually the universe, sometimes the Earth. Meaning the same as "S contains X" or "X fulfills the criteria for an element that define S". Henceforth we'll call S the reference set.
In order have a conclusive answer to a question of existence, the reference set should be explicitly mentioned with the question. For example, if someone says that gigantic fire-breathing flying lizards don't exist, they probably mean they don't exist in the set of creatures living on Earth. They wouldn't be so sure if the set was extended to the observable and unobservable universe. And they'd certainly be wrong if we were talking about the universes of fantasy literature.
So I take it that your question is actually "Does the universe exist in the universe?"
Or in other words: Is the universe a set that contains itself?
If a different reference set was implied, such as "the set of all possible alternate universes with similar physics to those observed by us", our universe does indeed exist.
What is the universe
The English wikipedia article contains these:
The Universe is all of time and space and its contents
The Universe can be defined as everything that exists, everything that has existed, and everything that will exist
Unfortunately, if (our) Universe is also the default implicit reference set, the second definition is circular and thus not very useful. Ultimately the universe depends on the observer. It's all the time, space and content in the structure that permeates their observations.
Just a thought. If nothing exists, but there's no reference either against which not to exist, it's not the same as nothing absolutely existing.
It's the relation of nothing to undefined. Like zero divided by zero (0/0), this can be anything, not just any number.