What the axioms mean…
But I cannot understand the inference results produced by Pellet reasoner. The restriction on the existence of the property seems not to work. How can I define such a class?
You haven't said what the inference results that Pellet produces are, so it's hard to explain them in particular. But, we can still take a look at what your axioms are saying. OWL classes are really like sets, and the restrictions that you're defining are making assertions about the elements of those sets. When you say, for instance, that
Base ⊑ ∃prop.String
What you're saying is that
1. If some x is a Base, then there is some String s such that prop(x,s).
Now, you have two more axioms:
Sub ⊑ Base
Sub ⊑ =0 prop.String
These say that
2. If some x is a Sub, then x is also a Base.
3. If some x is a Sub, then there are exactly zero Strings s where prop(x,s).
…and why Sub is a subclass of everything else.
Moreover could you please explain why the Sub class is inferred to be a subclass of ObjProp.
Your ontology isn't inconsistent, but you do have the typically undesirable state of affairs that one of your classes, Sub, can't have members. Why is this the case? Well, suppose some x is a Sub. Then, because x is a Sub, it has exactly zero strings. Of course, every Sub is also a Base, so x is also a Base. Since it's a Base, then it must have at least one string. It can't have zero and at least one, so we can't actually have any instances of Sub. Since Sub is the empty class (i.e., equivalent to owl:Nothing) it's a subclass of every other class.