Sharp weighted Trudinger-Moser and Caffarelli-Kohn-Nirenberg inequalities and their extremal functions

The main purpose of this paper is to establish sharp weighted Trudinger-Moser inequalities (Theorems 1.1, 1.2 and 1.3) and Caffarelli-Kohn-Nirenberg inequalities in the borderline case p=N (Theorems 1.5, 1.6 and 1.7) with best constants. Existence of extremal functions is also investigated for both the weighted Trudinger-Moser and Caffarelli-Kohn-Nirenberg inequalities. Radial symmetry of extremal functions for the weighted Trudinger-Moser inequalities are established (Theorem 1.4). Moreover, the sharp constants and the forms of the optimizers for the Caffarelli-Kohn-Nirenberg inequalities in some particular families of parameters in the borderline case p=N will be computed explicitly. Symmetrization arguments do not work in dealing with these weighted inequalities because of the presence of weights and the failure of the Polyá - Szegö inequality with weights. We will thus use a quasi-conformal mapping type transform and the corresponding symmetrization lemma to overcome this difficulty and carry out proofs of these results. As an application of the Caffarelli-Kohn-Nirenberg inequality, we also establish a weighted Moser-Onofri type inequality on the entire Euclidean space R2 (see Theorem 1.8).