1. Necessarily, everything that exists is either contingent or necessary. (Definition)
2. Necessarily, something exists. (Premise)
3. Possibly, nothing contingent exists. (Premise)
4. Therefore, something necessary exists. (From 1 - 3)
Reductio ad absurdum:
5. Nothing necessary does exists. (Assumption)
6. Necessarily, if nothing necessary exists, then it's possible for nothing to exist. (From 1 and 3)
7. (6) contradicts (2).
8. Therefore, (5) is false and something necessary exists.
The argument is logically airtight. If the premises are true, then there's no escaping the conclusion. Premises (1) and (3) are either true by definition or true on any realistic account of ontology. The key premise is (2). Is it necessarily the case that something exists? One could certainly defend cosmological arguments and the PSR, but I'd like to see another type of argument. Can it be shown that "possibly, nothing exists" (possibility of ontological nihilism) is contradictory or absurd?