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arXiv · 2003.09440

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

Marta Latorre·Francescantonio Oliva·Francesco Petitta·Sergio Segura de León
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Abstract

We study the Dirichlet problem for an elliptic equation involving the $1$-Laplace operator and a reaction term, namely: $$ \left\{\begin{array}{ll} \displaystyle -\Delta_1 u =h(u)f(x)&\hbox{in }\Omega\,,\\ u=0&\hbox{on }\partial\Omega\,, \end{array}\right. $$ 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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Marta Latorre, Francescantonio Oliva, Francesco Petitta, Sergio Segura de León. 2020-03-20. The Dirichlet problem for the $1$-Laplacian with a general singular term and $L^1$-data. https://doi.org/10.1088/1361-6544%2Fabc65b

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