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 $$\{\begin{array}{ll}-{\Delta }_{1}u=g(u)|Du|+h(u)f& \text{in}\Omega ,\\ u=0& \text{on}\partial \Omega ,\end{array}$$ where $\Omega$ is an open bounded subset of ${\mathbb{R}}^N$, $f\ge 0$ 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.

References in corpus (3)

• Existence and nonexistence of solutions for singular quadratic quasilinear equations
• The Dirichlet problem for singular elliptic equations with general nonlinearities
• The Dirichlet problem for the $1$-Laplacian with a general singular term and $L^1$-data

Cited by in corpus (1)

• Finite energy solutions for nonlinear elliptic equations with competing gradient, singular and $L^1$ terms