A 𝑝-th Yamabe equation on graph

Abstract

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