Best Constants and Maximizers of Caffarelli-Kohn-Nirenberg and Moser-Trudinger Inequalities

In this talk, we will describe some recent works on best constants and maximizers of Caffarelli-Kohn-Nirenberg inequalities with the presence of the weights. Best constants are identied and existence of maximizers are proved and in some special cases explicit forms of maximizers are found. Existence and Symmetry of maximizers of Moser-Trudinger inequalities are also discussed. These are joint works with Nguyen Lam (University of Pittsburgh), Mengxia Dong (Wayne State University), Lu Zhang (Wayne State University).