Existence and nonexistence of positive solutions of $p$-Kolmogorov equations perturbed by a Hardy potential

Jerome A. Goldstein, Daniel Hauer, Abdelaziz Rhandi

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 [HaRh2013]. Our results in this paper extend the ones in [GoRh2011] concerning linear Kolmogorov operators 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 $1<p<\mathrm{\infty }$, and secondly, by establishing nonexistence locally in time of positive weak solutions of such equations without using any growth conditions.

Keywords: weighted Hardy inequality, nonlinear Ornstein-Uhlenbeck operator, $p$-Laplace operator, singular perturbation, existence, nonexistence.

AMS Subject Classification: Primary MSC[2010]; secondary 35A01, 35B09, 35B25, 35D30, 35D35, 35K92.