1-Laplacian type problems with strongly singular nonlinearities and gradient terms

Abstract

We show optimal existence, nonexistence and regularity results for nonnegative solutions to Dirichlet problems as [Formula: see text] where [Formula: see text] is an open bounded subset of [Formula: see text], [Formula: see text] belongs to [Formula: see text], and [Formula: see text] and [Formula: see text] are continuous functions that may blow up at zero. As a noteworthy fact we show how a non-trivial interaction mechanism between the two nonlinearities [Formula: see text] and [Formula: see text] produces remarkable regularizing effects on the solutions. The sharpness of our main results is discussed through the use of appropriate explicit examples.