This project will consider semilinear partial differential equations with critical exponents. Topics to be studied include the Kazdan-Warner problem in bounded regions with Dirichlet boundary condition, further analysis of the constant in the Sobolev inequality, and multiplicity of solutions for the Neumann problem. These problems are motivated by questions arising in areas as diverse as differential geometry and mathematical biology. Basic questions such as the existence of surfaces with prescribed curvature or the properties of the equilibria of biological population models will benefit from the results of this project.
