I came to the following question when thinking about the (infinitely generated) tilting-cotilting correspondence, where it appears to be relevant.

Does there exist a locally presentable abelian category with enough injective objects that is not a Grothendieck category?

Background: an abelian category is called Grothendieck if it has a generator and satisfies the axiom Ab5. The latter means that (the category is cocomplete and) the filtered colimit functors are exact. Any locally presentable category has a generator, so the question is really about existence of locally presentable abelian categories with enough injective objects, but nonexact filtered colimits.

An abelian category is said to satisfy Ab3 if it is cocomplete. Any locally presentable category is cocomplete by definition. An abelian category is said to satisfy Ab4 if it is cocomplete and the coproduct functors are exact. Any cocomplete abelian category with enough injective objects satisfies Ab4.

The category opposite to the category of vector spaces over some fixed field is cocomplete and has enough injective objects, but it does not satisfy Ab5 (because the category of vector spaces does not satisfy Ab5*, i.e., does not have exact filtered limits). The category opposite to the category of vector spaces is not locally presentable, though.

Any locally finitely presentable abelian category is Grothendieck, and any Grothendieck abelian category is locally presentable, but the converse implications do not hold. Any Grothendieck abelian category has enough injective objects. Does a locally presentable abelian category with enough injective objects need to be Grothendieck?