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

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The Dirichlet problem for the 1-Laplacian with a general singular term and L 1-data
Nonlinearity ( IF 1.6 ) Pub Date : 2021-03-31 , DOI: 10.1088/1361-6544/abc65b
Marta Latorre ₁ , Francescantonio Oliva ₂ , Francesco Petitta ₃ , Sergio Segura de Len ₄

Affiliation

We study the Dirichlet problem for an elliptic equation involving the 1-Laplace operator and a reaction term, namely: $\begin{cases}-{{\Delta}}_{1}u=h\left(u\right)f\left(x\right)\hfill & \text{in}\;{\Omega},\hfill \\ u=0\hfill & \text{on}\;\partial {\Omega},\hfill \end{cases}$ where ${\Omega}\subset {\mathbb{R}}^{N}$ is an open bounded set having Lipschitz boundary, f ∈ L ¹(Ω) 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.

中文翻译：

具有一般奇异项和 L 1-数据的 1-拉普拉斯算子的狄利克雷问题

我们研究涉及 1-拉普拉斯算子和反应项的椭圆方程的狄利克雷问题，即： $\begin{cases}-{{\Delta}}_{1}u=h\left(u\right)f\left(x\right)\hfill & \text{in}\;{\Omega}, \hfill \\ u=0\hfill & \text{on}\;\partial {\Omega},\hfill \end{cases}$ 其中 ${\Omega}\subset {\mathbb{R}}^{N}$ 是具有 Lipschitz 边界的开有界集，f ∈ L ¹ (Ω) 是非负的，h是连续实函数可能会在零时爆炸。我们调查数据的最佳范围，以获得非负解的存在、不存在和（无论何时）唯一性。

更新日期：2021-03-31

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