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arXiv 2303.17232math.AP
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关于边界上带有吸收项的非线性Robin问题和$L^1$数据

On a nonlinear Robin problem with an absorption term on the boundary and $L^1$ data

Francesco Della Pietra, Francescantonio Oliva, Sergio Segura de León

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摘要：

本文讨论了\begin{equation*} \left\{ \begin{array}{l} -Δu = f(x) \text{ in }Ω, \frac{\partial u}{\partial ν} + λ(x) u = \frac{g(x)}{u^η} \text{ on } \partialΩ, \end{array} \right. \end{equation*}非负解的存在唯一性，其中$η\ge 0$，$f,λ$，$g$是非负可积函数.集合$Ω\subset\mathbb{R}^N (N> 2)$是光滑边界开有界的，$ν$表示其单位向外法向量.更一般地说，我们处理由$p$ -拉普拉斯型单调算子与非线性边界条件共同驱动的方程。我们证明了熵解的存在性，并在自然假设下检查该解是唯一的。除其他功能外，我们研究的正规化效果的解决方案的吸收和非线性边界项。

英文摘要：

We deal with existence and uniqueness of nonnegative solutions to \begin{equation*} \left\{ \begin{array}{l} -Δu = f(x) \text{ in }Ω, \frac{\partial u}{\partial ν} + λ(x) u = \frac{g(x)}{u^η} \text{ on } \partialΩ, \end{array} \right. \end{equation*} where $η\ge 0$ and $f,λ$ and $g$ are nonnegative integrable functions. The set $Ω\subset\mathbb{R}^N (N> 2)$ is open and bounded with smooth boundary and $ν$ denotes its unit outward normal vector. More generally, we handle equations driven by monotone operators of $p$-Laplacian type jointly with nonlinear boundary conditions. We prove existence of an entropy solution and check that this solution is unique under natural assumptions. Among other features, we study the regularizing effect given to the solution by both the absorption and the nonlinear boundary term.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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