In normal usage, only persists is possible. However, in certain registers, it is possible to use the subjunctive form persist in this situation, because it is the main verb in an if-clause.
The Cambridge Grammar of the English Language gives the following example on page 1000:
He struggles in vain against the proposition that if the mind be immaterial, its functions ought to be unaffected by the condition of the body.
The Grammar says that the subjunctive is "fairly rare [in conditionals], especially with verb-forms other than be; it belongs to formal style and verges on the archaic."
Here is an example with a verb other than be, taken from Higher Course Geometry by Forder (1930):
If the ray HA' cut the circle ABC in T, prove that HA' = A'T, TK || BC.
(It is clear from the context that the past tense is not intended here.)
In summary, you may occasionally encounter the subjunctive mood (persist) after if in highly formal texts, particularly older ones. However it is extremely unlikely that you will want to use anything other than the indicative mood (persists) in your own writing.