TITLE: Failure of GCH at a strongly compact cardinal
QUESTION [6 upvotes]: Does Con(ZFC+ there exists a strongly compact cardinal) imply Con(ZFC+ there exists a strongly compact cardinal $\kappa+ 2^\kappa > \kappa^+$)?

REPLY [4 votes]: The answer to the question is yes, as it is proved by Usuba in the paper
Strongly compact cardinals and the continuum function
.
In fact Usuba proves the following more general result:
Theorem. Assume $\kappa$ is a strongly compact cardinal. Then there exists a generic extension in which $\kappa$ remains strongly compact and indestructible under adding any number of new subsets to $\kappa$ (Theorem 3.1 of the above paper).