Singular solutions for coercive quasilinear elliptic inequalities with nonlocal terms. (August 2020)

• Record Type: Journal Article
• Authors: Filippucci, Roberta; Ghergu, Marius
• Is Part Of: Nonlinear analysis. Volume 197(2020)
• Publication Date: 2020-08
• Subjects: 35J62 -- 35A23 -- 35B09 -- 35B53; Quasilinear elliptic inequalities -- Weighted m-Laplace operator -- Singular 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.2020.111857
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study the inequality div ( | x | − α | ∇ u | m − 2 ∇ u ) ≥ ( I β ∗ u p ) u q in B 1 ∖ { 0 } ⊂ R N, where α > 0, N ≥ 1, m > 1, p, q > m − 1 and I β denotes the Riesz potential of order β ∈ ( 0, N ) . We obtain sharp conditions in terms of these parameters for which positive singular solutions exist. We further establish the asymptotic profile of singular solutions to the double inequality a ( I β ∗ u p ) u q ≥ div ( | x | − α | ∇ u | m − 2 ∇ u ) ≥ b ( I β ∗ u p ) u q in B 1 ∖ { 0 } ⊂ R N, where a ≥ b > 0 are constants.