If something exists, it is actually present in the world.
It is clear that a serious problem exists.
They walked through my office as if I didn't exist.
When exist has this meaning, don't use it in a progressive form. Don't say, for example, ‘It is clear that a serious problem is existing’.
You also use exist to say that someone manages to live under difficult conditions or with very little food or money.
How can we exist out here?
The whole band exist on a diet of chocolate and crisps.
When exist has this meaning, it can be used in a progressive form.
People were existing on a hundred grams of bread a day.