Field equations with cosmological constant Is it enough to add the cosmological constant in the einstein field equations to guarantee that the obtained solution describes an expanding universe?
 A: I believe you’re asking if adding a cosmological constant term would guarantee expansion in any solution to the EFE with it, and not just if it’s necessary for expansion, but i’ll explain this anyway just to be clear.
You can find a solution for an expanding universe without a cosmological constant in the first place. All you have to do is make the stress energy-tensor equal to the mass-energy density of the universe. You can then find a general form of the metric, for say, a flat expanding universe (this is what i did because it is experimentally observed to be flat as far as we know). After that you solve the einstein field equations to get the friedmann equations which you can then solve for something called the scale factor, $a(t)$, which is a function of time which determines the scale of space itself. 
The mass-energy density is a bit tricky though because the term for that changes depending on whether or not the universe is filled with mostly matter or radiation. Earlier in the universe it was filled with radiation so doing a matter-filled approximation will give you the wrong answer for the age of the universe, something around 9 billion years. So there’s more to take into account.
You can solve the Einstein Field Equations for an expanding universe with no cosmological constant which are $$R_{μν}- \frac{1}{2}g_{μν}R = \frac{8πG}{c^4}T_{μν}$$
Where T_μν is the stress-energy tensor i was talking about. 
The cosmological constant was added by einstein because of his equations predicting a changing universe, which he didn’t like. Like many during his time, Einstein was a proponent of “Steady state” theory, which was that the universe was static and unchanging, not expanding. The expanding universe is caused by the mass-energy density of it. 
Nowadays the cosmological constant is also used to explain how the expansion of the universe is accelerating. 
As the universe expands, the density of energy should change right? Because things are getting further apart. Well dark energy is essentially a constant energy density that stays the same regardless of the expansion, which is pretty weird but it’s called dark energy for a reason.
But to answer your question, i haven’t worked very much with the cosmological constant, but since an energy density of the universe is a prerequisite to an expanding universe, adding a cosmological constant which is a constant energy density to the EFE seems to make it inevitable that it would at least induce some sort of expansion (depending on the value of the cosmological constant). This is the case with the “De-Sitter Schwarszchild Metric” which describes a pseudo-schwarschild spacetime that’s inflating. The De-Sitter Schwarszschild metric is the simplest solution to the EFE with a cosmological constant.
Apologies for rambling but i want to be as specific as possible i guess.
EDIT: The solutions OP is asking for do exist. They are called Lambdavacuum solutions: https://en.m.wikipedia.org/wiki/Lambdavacuum_solution
So to anybody reading, yes the cosmological constant alone will induce expansion.
