lust after vs want

lust after is not a noun while want is a noun.

Word	Noun	Adjective	Verb	Adverb
lust after	No	No	Yes	No
want	Yes	No	Yes	No

As verbs, want is a hypernym of lust after; that is, want is a word with a broader meaning than lust after:

Other hypernyms of lust after include desire.

lust after (noun)	want (noun)
	a specific feeling of desire
	anything that is necessary but lacking
	a state of extreme poverty
	the state of needing something that is absent or unavailable

lust after (verb)	want (verb)
have a strong sexual desire for	wish or demand the presence of
	have need of
	hunt or look for; want for a particular reason
	feel or have a desire for; want strongly
	be without, lack; be deficient in