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?

**EDIT**: In response to one of the comments, let me add that the thematic example of a locally presentable abelian category that is not locally finitely presentable and, moreover, not Grothendieck is the category of Ext-$p$-complete (weakly $p$-complete) abelian groups, as mentioned in Martin Frankland's answer to this question -- What's an example of a locally presentable category "in nature" that's not $\aleph_0$-locally presentable? . This category is not Grothendieck because the colimit of the sequence of monomorphisms $\mathbb Z_p\overset{p}\longrightarrow \mathbb Z_p\overset{p}\longrightarrow\mathbb Z_p\overset{p}
\longrightarrow\dotsb$ vanishes in it (hence filtered colimits do not preserve monomorphisms and do not commute with the kernels).

The Ext-$p$-complete abelian groups are otherwise known as "$p$-contramodule $\mathbb Z$-modules" or "contramodules over the topological ring of $p$-adic integers $\mathbb Z_p$". Generally, various contramodule categories provide examples of locally presentable (often, but not always, locally $\aleph_1$-presentable) abelian categories which are not locally finitely presentable, not Grothendieck, and, typically, have enough projective objects but no nonzero injectives.

Let me cite our paper with Jiří Rosický "Covers, envelopes, and cotorsion theories in locally presentable abelian categories and contramodule categories", Journ. of Algebra 483 (2017), arXiv version at https://arxiv.org/abs/1512.08119 , and my preprint "Abelian right perpendicular subcategories in module categories", https://arxiv.org/abs/1705.04960 , as references on locally presentable abelian categories. In particular, according to Example 4.1(4) in the former paper, there are no nonzero injective objects in the category of Ext-$p$-complete abelian groups.