Existence and nonexistence of positive solutions of [formula omitted]-Kolmogorov equations perturbed by a Hardy potential

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.