Sharp weighted Trudinger–Moser and Caffarelli–Kohn–Nirenberg inequalities and their extremal functions. (August 2018)

• Record Type: Journal Article
• Authors: Dong, Mengxia; Lam, Nguyen; Lu, Guozhen
• Is Part Of: Nonlinear analysis. Volume 173(2018)
• Page Start: 75
• Page End: 98
• Publication Date: 2018-08
• Subjects: Caffarelli–Kohn–Nirenberg inequality -- Moser–Onofri inequality -- Weighted Trudinger–Moser inequalities -- Best constants -- Extremal functions -- Quasi-conformal transforms; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2018.03.006
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: 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 R 2 (see Theorem 1.8).