A nonlinear biharmonic equation on hypergraphs
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Inspired by the foundational work of Shao, Tian, and Zhao \cite{r1} about hypergraphs, this paper aims to explore the biharmonic equation and expand application range of biharmonic equation. We focus on a connected finite hypergraph $H=(V, E)$, which generalizes traditional graphs by allowing edges to connect more than two vertices. Our investigation centers on the biharmonic equation \begin{equation*} \Delta^{2} \phi-\mathrm{div}(x(v)\nabla \phi)+y(v)\phi=\frac{f(v, \phi)}{\delta_{e}^{\alpha}}+\beta g(v), \end{equation*} where $x(v)$ and $y(v)$ are two functions defined on $V$, $\alpha \ge0, \beta >0$. Here, $\delta_{e}$ denotes the degree of hyperedge $e\in E$, defined as $\delta_{e}=|e|$, which represents the cardinality of the vertices contained in $e$. Under certain conditions on the functions $f$ and $g$, we establish the existence of two distinct positive solutions to the biharmonic equation.
Abstract
Inspired by the foundational work of Shao, Tian, and Zhao \cite{r1} about hypergraphs, this paper aims to explore the biharmonic equation and expand application range of biharmonic equation. We focus on a connected finite hypergraph $H=(V, E)$, which generalizes traditional graphs by allowing edges to connect more than two vertices. Our investigation centers on the biharmonic equation \begin{equation*} \Delta^{2} \phi-\mathrm{div}(x(v)\nabla \phi)+y(v)\phi=\frac{f(v, \phi)}{\delta_{e}^{\alpha}}+\beta g(v), \end{equation*} where $x(v)$ and $y(v)$ are two functions defined on $V$, $\alpha \ge0, \beta >0$. Here, $\delta_{e}$ denotes the degree of hyperedge $e\in E$, defined as $\delta_{e}=|e|$, which represents the cardinality of the vertices contained in $e$. Under certain conditions on the functions $f$ and $g$, we establish the existence of two distinct positive solutions to the biharmonic equation.