Are there any known weaknesses with ElGamal algorithm?

I am working on a messaging system and plan to use ElGamal asymmetric encryption to protect the message contents during storage and transmission. (This would be in addition to TLS used during transmission.) What I want to know is if there are any inherent weaknesses in the ElGamal algorithm that I should be aware of or guidelines to the key bit size.

Assumptions:

1. Messages will be short, so performance is less of a concern.


2. Keys (and the associated user accounts) are disposable (short lived).


3. Message retention will be minimized (days, not weeks or months).


4. Messaging will be asynchronous as users are not expected to be online at the same time. (This is the main reason that I have ruled out the typical D-H key exchange to generate a shared, symmetric key.)


5. Private keys will never leave the client device.


6. Public keys will be stored in a database with the anonymous user account consisting of a username and hashed password.