fail to satisfy = frustrate?
Solution 1:
Among the definitions of to frustrate that appear in the O.E.D., the one that seems to be the most relevant here is ‘to balk, disappoint (a hope, expectation, etc.)’. The ‘etc.’ in the definition may be taken to cover needs, so it does not seem unreasonable to speak of one’s needs being frustrated, and indeed a casual search will readily reveal examples of such constructions. One could use to frustrate a need as the opposite of to satisfy a need, but only in certain contexts. The phrase to frustrate a need is not always interchangeable with to not satisfy a need, for the following reasons.
(1) The word to frustrate would typically be understood to imply that some active effort is involved, that we are not dealing with a mere failure to act. If pure laziness results in some need not being satisfied, that would not normally be described as frustrating the need.
(2) One can speak of satisfying one’s own needs and of satisfying the needs of others. On the other hand, while one can frustrate the needs of others, it would be very odd to speak of frustrating one’s own needs. That is probably a consequence of (1): absent some highly unusual circumstances, people don’t actively work on preventing the satisfaction of their own needs.
(3) To speak of somebody frustrating somebody’s need is apt only if there is some reasonable expectation that one would satisfy it.
(4) While the verb to frustrate can take need (as well as hope, expectation, desire, want, etc.) as its object, it nowadays much more often takes some noun standing for a person as its object.