Maximizers for Fractional Caffarelli–Kohn–Nirenberg and Trudinger–Moser Inequalities on the Fractional Sobolev Spaces

This paper is mainly concerned with the existence of extremals for the fractional Caffarelli–Kohn–Nirenberg and Trudinger–Moser inequalities. We first establish the existence of extremals of the fractional Caffarelli–Kohn–Nirenberg inequalities in Theorem 1.1. It is also proved that the extremals of this inequality are the ground-state solutions of some fractional p-Laplacian equation. Then, we use the method of considering the level-sets of functions under consideration first developed by Lam and Lu (Adv Math 231(6):3259–3287, 2012; J Differ Equ 255(3), 298–325, 2013) and combining with a new compactness argument and the fractional rearrangement inequalities to establish the existence of extremals for the fractional Trudinger–Moser inequalities with the Dirichlet norm (see Theorems 1.2, 1.3, 1.6 and 1.7). Radial symmetry of extremals and concentration-compactness principle for the fractional Trudinger–Moser inequalities are also established in Theorems 1.4, 1.8 and 1.9. Finally, we investigate some relationship between the best constants of the fractional Trudinger–Moser and Caffarelli–Kohn–Nirenberg inequalities in the asymptotic sense (see Theorem 1.11).