The Dirichlet problem for the $1$-Laplacian with a general singular term and $L^1$-data

Abstract

We study the Dirichlet problem for an elliptic equation involving the $1$-Laplace operator and a reaction term, namely: $$\{\begin{array}{ll}-{\Delta }_{1}u=h(u)f(x)& in\; \Omega ,\\ u=0& on\partial \Omega ,\end{array}$$ where $\Omega \subset {\mathbb{R}}^N$ is an open bounded set having Lipschitz boundary, $f\in L^1(\Omega )$ is nonnegative, and $h$ is a continuous real function that may possibly blow up at zero. We investigate optimal ranges for the data in order to obtain existence, nonexistence and (whenever expected) uniqueness of nonnegative solutions.

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