How to start doing undergraduate research in mathematics when I have limited support from my professors?
I think this can only be explained with a personal story. I did my undergraduate at a small liberal arts institution and because of that there were not many professors to choose from. However, I knew who I wanted to work with since his general area (dynamical systems) always interested me, although we had never met (since he was on sabbatical for awhile).

I sent him an email listing 3 different projects I had thought of, and he emailed me back saying he wouldn't do those, but he had a project I should do (climate mathematics, never even thought about it before). I was kind of taken aback, but in the end I thought, well this is the best I am getting, and took the project.

When we finally met the following year, I was still a little skeptical, but as we started working through the problems I found out how interesting the subject was. I learned a lot through his mentorship, and gave chalkboard talks on different papers / analysis results (working out of Courant/Hilbert and Wittaker) every few weeks and a few months in I was very fond of the project and was spending all of my time on it just because of how interesting it was, and the story ends with it turning into a good publication.

Turns out the professor had the idea planned out during sabbatical. He was working with a research team there where many ideas were created, and the papers I worked from were recent results by newly minted PhDs he was working with. He had even published a paper on how to teach undergraduates climate science: this man had though it through!

There are a few morals to the story. First of all, when you're young in your research career, you don't know what is interesting yet, and a quick read of the literature is not enough to gauge this. Also, if plenty of people are researching a subject, it's probably pretty interesting and worth a try. Lastly, for research with undergraduates, the good mentors would like you to go with their ideas because they have a plan for how to teach it / work through the problem. 

All together, I would say find a mentor you respect and trust and just ask them to take you under their wing. It may not be the project you had in mind, but you will learn a lot more this way, and you'll be surprised how interesting the project is.