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

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Abstract

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).

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Acknowledgements

Lu Chen was partly supported by a grant from the National Natural Science Foundation of China (No. 11901031) and a grant from Beijing Institute of Technology (No. 3170012221903). G. Lu was partly supported by a Simons Collaboration Grant from the Simons Foundation. The authors wish to thank the referee for his very careful reading and useful comments.

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Authors and Affiliations

  1. School of Mathematical and statistics, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China

    Lu Chen

  2. Department of Mathematics, University of Connecticut, Storrs, CT, 06269, USA

    Guozhen Lu

  3. Department of Applied Mathematics, School of Mathematics and Physics, Beijing University of Science and Technology, Beijing, 100083, China

    Caifeng Zhang

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  1. Lu Chen
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  2. Guozhen Lu
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  3. Caifeng Zhang
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Chen, L., Lu, G. & Zhang, C. Maximizers for Fractional Caffarelli–Kohn–Nirenberg and Trudinger–Moser Inequalities on the Fractional Sobolev Spaces. J Geom Anal 31, 3556–3582 (2021). https://doi.org/10.1007/s12220-020-00406-1

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