Existence and nonexistence of positive solutions of p-Kolmogorov equations perturbed by a Hardy potential. (January 2016)

• Record Type: Journal Article
• Authors: Goldstein, Jerome A.; Hauer, Daniel; Rhandi, Abdelaziz
• Is Part Of: Nonlinear analysis. Volume 131(2016)
• Page Start: 121
• Page End: 154
• Publication Date: 2016-01
• Subjects: 35A01 -- 35B09 -- 35B25 -- 35D30 -- 35D35 -- 35K92; Weighted Hardy inequality -- Nonlinear Ornstein–Uhlenbeck operator -- p-Laplace operator -- Singular perturbation -- Existence -- Nonexistence; 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.2015.07.016
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this article, we establish the phenomenon of existence and nonexistence of positive weak solutions of parabolic quasi-linear equations perturbed by a singular Hardy potential on the whole Euclidean space depending on the controllability of the given singular potential. To control the singular potential we use a weighted Hardy inequality with an optimal constant, which was recently discovered in Hauer and Rhandi (2013). Our results in this paper extend the ones in Goldstein et al. (2012) concerning a linear Kolmogorov operator significantly in several ways: firstly, by establishing existence of positive global solutions of singular parabolic equations involving nonlinear operators of p -Laplace type with a nonlinear convection term for 1 < p < ∞, and secondly, by establishing nonexistence locally in time of positive weak solutions of such equations without using any growth conditions.