具有 Hardy 势的奇异 $1$-Laplace 方程的尖锐存在性与不存在性

中文摘要 AI 辅助

本文研究奇异 Dirichlet 问题 \begin{equation}\label{abs1} \left\{ \begin{array}{rclr} -\Delta_1u &=&\dfrac{\lambda}{|x|}\hbox{Sgn}\\,(u)+\dfrac{f}{u^\gamma}&\quad\mbox{in}\\; \Omega,\\\\[1ex] u &=&0& \quad\mbox{on}\\; \partial\Omega, \end{array} \right. \end{equation} 其中 $\Delta_1u=\hbox{div}\left(\frac{Du}{|Du|}\right)$ 表示 $1$-Laplace 算子，参数为 $\lambda\in {R}$ 和 $\gamma>0$，$f$ 是属于 Lorentz 空间 $L^{N,\infty}(\Omega)$ 的非负函数。我们的主要目标是在尖锐限制 $\lambda<N-1$ 下建立非平凡解的存在性，无论数据 $f$ 的大小或奇异指数 $\gamma>0$ 的值如何。我们还证明这些解是全局有界的，并且相反地，一旦 $\lambda\geq N-1$ 且 $f$ 为正，则不存在解。这些结果通过严格分析当 $p\to 1^+$ 时逼近 $p$-Laplace 问题 \begin{equation} \left\{ \begin{array}{rclc} -\Delta_pu&=&\dfrac{\lambda}{|x|^p}|u|^{p-2}u+\dfrac{f}{u^\gamma}&\quad\mbox{ in }\\; \Omega,\\\\[1ex] u&=&0&\quad\mbox{ on }\\; \partial\Omega. \end{array} \right. \end{equation} 的解的渐近行为获得。最后，我们提供一族显式例子，旨在说明我们主要假设的尖锐最优性。

英文摘要

In this paper, we investigate the singular Dirichlet problem \begin{equation}\label{abs1} \left\{ \begin{array}{rclr} -Δ_1u &=&\dfracλ{|x|}\hbox{Sgn}\,(u)+\dfrac{f}{u^γ}&\quad\mbox{in}\; Ω,\\[1ex] u &=&0& \quad\mbox{on}\; \partialΩ, \end{array} \right. \end{equation} where $Δ_1u=\hbox{div}\left(\frac{Du}{|Du|}\right)$ denotes the $1$-Laplacian operator, the parameters are $λ\in {R}$ and $γ>0$, and $f$ is a non-negative function belonging to the Lorentz space $L^{N,\infty}(Ω)$. Our main goal is to establish the existence of non-trivial solutions under the sharp restriction $λ<N-1$, regardless of the magnitude of the datum $f$ or the value of the singular exponent $γ>0$. We also show that these solutions are globally bounded and that, on the contrary, no solution exists as soon as $λ\geq N-1$ and $f$ is positive. These results are achieved by rigorously analyzing the asymptotic behavior, as $p\to 1^+$, of the solutions to the approximating $p$-Laplace problems \begin{equation} \left\{ \begin{array}{rclc} -Δ_pu&=&\dfracλ{|x|^p}|u|^{p-2}u+\dfrac{f}{u^γ}&\quad\mbox{ in }\; Ω,\\[1ex] u&=&0&\quad\mbox{ on }\; \partialΩ. \end{array} \right. \end{equation} Finally, we provide a family of explicit examples designed to illustrate the sharp optimality of our main assumptions.

发表机构

• Universidade Federal de São Carlos（圣卡洛斯联邦大学）
• Sapienza Università di Roma（罗马第一大学）

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