Research in these fields might be out of reach for an undergrad. These are mature fields and most of the current work in what you’ve mentioned will either be too specialized or too abstract.
For example, there is a theory for Sobolev spaces on sets that don’t have a Euclidean structure, e.g. metric measure spaces (you cannot “add” points or “scale” them like you can for points in Rn). There are quite a lot of steps to go through from someone starting to learn about Sobolev spaces for subsets of Rn to this.
With regards to well-posedness, researchers would often prove it for a specific model that is not quite the same as the ones you see in classical literature. For example, you might consider models coming from the sciences with very specific terms that, because of strong connections to either physics, chemistry, or biology, might say something more about the solution than what classical theory will tell you.
There are plenty more to look at for elliptic PDEs aside from existence and uniqueness. Mathematicians are also interested about the regularity/smoothness of their solutions. These depend on the structure of the equations you’re studying, the regularity of data and the coefficients of your equations, and the regularity of the domain boundary (if you’re working on bounded domains or domains with boundaries). There is also an entirely different approach for fully nonlinear elliptic PDEs that do not use Sobolev spaces. We call them viscosity solutions, and there is a jungle of problems and research directions to go towards to.
So going back to your original post, research in the sense of contributing something new and publishing these results, is most likely out of reach for most undergrads.
What you can do as a thesis, and ask your department about this, is do an expository paper on these topics. The theory of Sobolev spaces on Rn alone is a lot for an undergrad to learn and write about.
Hope that helps. Good luck! Feel free to dm me if you have questions :)