## D. Giachetti - F. Oliva - F. Petitta

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

created by petitta on 30 Oct 2019

modified on 23 Sep 2021

[

BibTeX]

*Accepted Paper*

**Inserted:** 30 oct 2019

**Last Updated:** 23 sep 2021

**Journal:** Comm. Contemp. Math

**Year:** 2021

**Abstract:**

We show optimal existence, nonexistence and regularity results for
nonnegative solutions to Dirichlet problems as $$ \begin{cases} \displaystyle
-\Delta_{1} u = g(u)

D u

+h(u)f & \text{in}\;\Omega,\\ u=0 &
\text{on}\;\partial\Omega, \end{cases} $$ where $\Omega$ is an open bounded
subset of $\mathbb{R}^N$, $f\geq 0$ belongs to $L^N(\Omega)$, and $g$ and $h$
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 $g$ and
$h$ produces remarkable regularizing effects on the solutions. The sharpness of
our main results is discussed through the use of appropriate explicit examples.