A 𝑝-th Yamabe equation on graph

Ge, Huabin

doi:10.1090/proc/13929

A $p$-th Yamabe equation on graph
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by Huabin Ge

Proc. Amer. Math. Soc. 146 (2018), 2219-2224

DOI: https://doi.org/10.1090/proc/13929

Published electronically: January 8, 2018

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Abstract:

Assume $\alpha \geq p>1$. Consider the following $p$-th Yamabe equation on a connected finite graph $G$: \begin{equation*} \Delta _p\varphi +h\varphi ^{p-1}=\lambda f\varphi ^{\alpha -1}, \end{equation*} where $\Delta _p$ is the discrete $p$-Laplacian, $h$ and $f>0$ are known real functions defined on all vertices. We show that the above equation always has a positive solution $\varphi$ for some constant $\lambda \in \mathbb {R}$.

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