The Brezis–Nirenberg problem for the Laplacian with a singular drift in Rn and Sn. (July 2017)

• Record Type: Journal Article
• Authors: Benguria, Rafael D.; Benguria, Soledad
• Is Part Of: Nonlinear analysis. Volume 157(2017)
• Page Start: 189
• Page End: 211
• Publication Date: 2017-07
• Subjects: 35XX -- 35B33 -- 35A24 -- 35J25 -- 35J60; Brezis–Nirenberg problem -- Spaces of constant curvature -- Drift Laplacian -- Critical exponent -- Rellich–Pohozaev identity; 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.2017.03.006
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider the Brezis–Nirenberg problem for the Laplacian with a singular drift for a (geodesic) ball in both R n and S n, 3 ≤ n ≤ 5 . The singular drift we consider derives from a potential which is symmetric around the center of the (geodesic) ball. Here the potential is given by a parameter ( δ say) times the logarithm of the distance to the center of the ball. In both cases we determine the exact region in the parameter space for which positive smooth solutions of this problem exist and the exact region for which there are no solutions. The parameter space is characterized by the (geodesic) radius of the ball, δ, and λ, the coupling constant of the linear term of the Brezis–Nirenberg problem.