It turns out that these actually depend a bit on how you conceptualize "the internet" and where it is and how you access it! And different people have somewhat different conceptions of that so we end up with language that's not in practice standardized across common usage.
You're treading an area where prepositions are a bit fuzzy in making the transition from physical to virtual senses, and multiple versions can sound OK (or not). In this case, since I think you're trying to say "I can't seem to access the internet using the browser on my phone, but I can access it using some apps", I might formulate it more succinctly like:
The mobile internet doesn't work in any of my browsers, but does work in some apps.
It also sounds OK to me to talk about the mobile internet on a browser (or in, or within, or using, a browser), but sounds less OK to talk about the mobile internet on an app. But I don't know that I'd describe those alternatives as "incorrect."