How do I find sponsors for maintaining a mobile application?

Wow. I'm not personally an experienced entrepreneur, so I can't really coach you on specifics. But, I have some thoughts - but more info is needed, too! First of all, is your application already on the Google Play and Apple App stores, and is it getting downloads and users? If the answer is "Yes" and traction is evident, then you have some real thinking to do!

1. Which is more important to you at this time? Finishing your college studies or putting your formal education on hold to pursue your passion? Unless you have high confidence and ambition to pursue your app dream, I'd advise sticking with finishing your studies as "Priority One" (I'm assuming you've already invested significant time and money into college as is).

2. Is financing an issue for continuing your college studies as is? It's unclear from your description if that remains an issue or not. If it is, then visit the Financial Assistance Office at your college to see what options are available to you. While it's great if you can avoid taking out student loans, discuss with them if perhaps that's a workable option for you, too. Loans are not intrinsically "evil", provided you've sized up the outlay and have a plan to pay it back ahead. Since you're already productive at your passion, it sounds to me like you will be able to have career income to make loan payments, if it's necessary to take one to complete your formal education.

3. If you decide to put studies on hold and go "the start-up route", then you must "shop out" your application to those who can help finance and develop it for further growth. See if there is a Hacker Dojo or Tech Entrepreneur meet-up (check meetup.com) nearby; you can drop in there, talk with other aspiring entrepreneurs, and possibly make useful connections to angel investors or potential business partners. You can also try approaching Venture Capital offices directly with your ideas, app, and "elevator pitch" (does "Shark Tank" ring a bell?) As you can undoubtedly already sense, this will take time, energy, focus, and the ability to keep yourself financially supported in the meantime, but if you have great ideas, a great app, and can collaborate and negotiate with other professionals, then this may be an achievable option. So, not something to be taken lightly, but other entrepreneurs have gone this route and succeeded at it. Check with your college or professors and see if there's a counseling office who can provide guidance or a prof who can mentor you in this approach.

Finally, "Good Luck!" I think it's exciting that you're already at this point, it sounds like you have talent and the ability to deliver - both will take you far if you learn and apply the skills for successful entrepreneurship.