论文标题
涉及1-Laplacian操作员的Gelfand类型问题
Gelfand type problems involving the 1-Laplacian operator
论文作者
论文摘要
在本文中，Gelfand问题的理论适合1-拉普拉斯环境。具体而言，我们处理以下问题\ begin {equation*} \ left \ {\ begin {array} {cc} -Δ_1U=λf（u）＆\ hbox {in}ω\，; \\ [2mm] u = 0＆\ hbox {on} \partialΩ\;; \ end {array} \ right。 \end{equation*} where $Ω\subset\mathbb{R}^N$ ($N\ge1$) is a domain, $λ\geq 0$ and $f\>:\>[0,+\infty[\to]0,+\infty[$ is any continuous increasing and unbounded function with $f(0)>0$. 证明存在阈值$λ^*= \ frac {h（ω）} {f（f（0）} $（为$ h（ω）$，$ω$的cheeger常数为$ω$），当$λ>λ^*$始终是$λ^$的解决方案时，没有解决方案。对径向案例进行了更详细的分析，以表明存在多个解决方案（甚至是单数）以及涉及$ p $ - laplacian的问题的行为，因为$ p $ to $ p $趋向于1，这使我们能够通过额外的条件识别适当的解决方案。
In this paper, the theory of Gelfand problems is adapted to the 1--Laplacian setting. Concretely, we deal with the following problem \begin{equation*} \left\{\begin{array}{cc} -Δ_1u=λf(u) &\hbox{in }Ω\,;\\[2mm] u=0 &\hbox{on }\partialΩ\,; \end{array} \right. \end{equation*} where $Ω\subset\mathbb{R}^N$ ($N\ge1$) is a domain, $λ\geq 0$ and $f\>:\>[0,+\infty[\to]0,+\infty[$ is any continuous increasing and unbounded function with $f(0)>0$. It is proved the existence of a threshold $λ^*=\frac{h(Ω)}{f(0)}$ (being $h(Ω)$ the Cheeger constant of $Ω$) such that there exists no solution when $λ>λ^*$ and the trivial function is always a solution when $λ\leλ^*$. The radial case is analyzed in more detail showing the existence of multiple solutions (even singular) as well as the behaviour of solutions to problems involving the $p$--Laplacian as $p$ tends to 1, which allows us to identify proper solutions through an extra condition.
