Concentration-Compactness principle for Trudinger–Moser inequalities with logarithmic weights and their applications. (August 2020)

• Record Type: Journal Article
• Authors: Zhang, Caifeng
• Is Part Of: Nonlinear analysis. Volume 197(2020)
• Publication Date: 2020-08
• Subjects: 35J60 -- 35B33 -- 46E30; Concentration-compactness principle -- Trudinger–Moser inequality -- Weighted Sobolev spaces -- Ground state solutions; 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.2020.111845
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we establish a sharp concentration-compactness principle associated with the Trudinger–Moser inequality on Sobolev spaces with logarithmic weights. As applications, we establish the existence of ground state solutions to the following equation with critical double exponential nonlinearity − d i v ( | ∇ u | N − 2 ∇ u ν ( x ) ) = f ( x, u ), in B 1 ( 0 ), u > 0, in B 1 ( 0 ), u = 0, on ∂ B 1 ( 0 ) . where ν ( x ) = | log ( e | x | ) | N − 1 is the logarithmic weight, the nonlinear term f ( x, u ) is continuous, radial in x ∈ B 1 ( 0 ) and has critical double exponential growth which behaves like exp ( e e N u N ′ ) .