Date of Award
12-1-2024
Degree Name
Doctor of Philosophy
Department
Mathematics
First Advisor
Gluck, Mathew
Abstract
We establish the existence of infinitely many entire solutions to a conformally invariant partial differential equation on R^n with a nonlocal nonlinearity term. Under suitable assumptions on the dimension and the nonlocality, we show that these solutions must be sign-changing. The primary obstacle in finding a solution lies in the fact that standard variational methods do not apply easily due to the lack of compactness, which results from invariance under certain transformations, such as dilations. Our approach involves using stereographic projection to lift our problem to an equivalent problem on the unit sphere. By leveraging the symmetries of the sphere, we are able to overcome the lack of compactness.
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