Brouwer degree for mean field equation on graph
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Let $u$ be a function on a connected finite graph $G=(V, E)$. We consider the mean field equation \begin{equation}\label{5} -\Delta u=\rho\bigg{(}\frac{he^u}{\int_V he^ud\mu}-\frac{1}{|V|}\bigg{)}, \end{equation} where $\Delta$ is $\mu$-Laplacian on the graph, $\rho\in \mathbb{R}\backslash\{0\}$, $h: V\ra\mathbb{R^+}$ is a function satisfying $\min_{x\in V}h(x)>0$. Following Sun and Wang \cite{S-w}, we use the method of Brouwer degree to prove the existence of solutions to the mean field equation $(\ref{5})$. Firstly, we prove the compactness result and conclude that every solution to the equation $(\ref{5})$ is uniformly bounded. Then the Brouwer degree can be well defined. Secondly, we calculate the Brouwer degree for the equation $(\ref{5})$, say \begin{equation*}d_{\rho,h}=\left\{\begin{array}{lll} -1,\quad \rho>0,\\ \ 1,\quad \ \rho<0.\end{array}\ri. \end{equation*} Consequently, the equation (\ref{5}) has at least one solution due to the \linebreak Brouwer degree $d_{\rho,h}\neq0$.
Abstract
Let $u$ be a function on a connected finite graph $G=(V, E)$. We consider the mean field equation \begin{equation}\label{5} -\Delta u=\rho\bigg{(}\frac{he^u}{\int_V he^ud\mu}-\frac{1}{|V|}\bigg{)}, \end{equation} where $\Delta$ is $\mu$-Laplacian on the graph, $\rho\in \mathbb{R}\backslash\{0\}$, $h: V\ra\mathbb{R^+}$ is a function satisfying $\min_{x\in V}h(x)>0$. Following Sun and Wang \cite{S-w}, we use the method of Brouwer degree to prove the existence of solutions to the mean field equation $(\ref{5})$. Firstly, we prove the compactness result and conclude that every solution to the equation $(\ref{5})$ is uniformly bounded. Then the Brouwer degree can be well defined. Secondly, we calculate the Brouwer degree for the equation $(\ref{5})$, say \begin{equation*}d_{\rho,h}=\left\{\begin{array}{lll} -1,\quad \rho>0,\\ \ 1,\quad \ \rho<0.\end{array}\ri. \end{equation*} Consequently, the equation (\ref{5}) has at least one solution due to the \linebreak Brouwer degree $d_{\rho,h}\neq0$.
참고문헌 (16)
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