I want to say that something is trying to attain widespread use.
However, when I say widespread use, I really want to say that it is trying to be used by a few people (not necessarily that widespread).
Is there a word, or an expression for this?
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Maybe you could use “gaining traction” (which implies that it is getting more and more users, though maybe not widespread yet), or common, broadly used, or well adopted.
9 votes for this short answer, I feel you deserve more! So here’s some addition information:
In “gaining traction”, the word traction is used figuratively, as “motive power provided for the action of drawing a thing over a surface”. In this use, it is thus synonymous with momentum.
Maybe Gain widespread use within a group / community?
For example "My website has widespread use within the UK Airsoft Community" Whist the UK Airsoft community is only a fraction of a percent of the entire population and almost everyone you ask will never have heard of the site, it is still in widespread use within the specified group.
I think this really depends on what is being used and who is using it.
It sounds like mainstream would fit. Something that is mainstream is acceptable to most people, but not necessarily adopted by many. Basically something is mainstream if when its use is not considered odd, when it is not specific to a fringe group.
To be clear, you might want to separate the thing from the producers or promoters of the thing. It is the company behind the toothpaste that is pushing for wider adoption of the product, rather than the thing itself "trying to attain widespread use" .
Another way to say this is a company striving for great market-share, or pushing for increased use among such-and-such demographic, for instance.
If your case is less about market-share, you might use [the people behind ProjectX are trying to] increase its user base, to widen adoption, promote its use, evangelize the product[icky marketing speak], spread the word about the product, etc..