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

TL;DR

This paper investigates the existence, nonexistence, and regularity of solutions to a class of 1-Laplacian problems with singular nonlinearities and gradient terms, revealing how interactions between nonlinearities can enhance regularity.

Contribution

It provides new optimal results on existence and regularity for 1-Laplacian problems with singular nonlinearities and gradient terms, highlighting the regularizing effects of their interaction.

Findings

Interaction between nonlinearities g and h improves regularity.

Established sharp conditions for existence and nonexistence.

Provided explicit examples illustrating the results.

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.