A p-th Yamabe equations on graph

Assume $α\geq p>1$. Consider the following $p$-th Yamabe equation on a connected finite graph $G$: $Δ_pφ+hφ^{p-1}=λfφ^{α-1},$ where $Δ_p$ is the discrete $p$-Laplacian, $h$ and $f>0$ are fixed real functions defined on all vertices. We show that the above equation always has a positive solution $φ$ for some constant $λ\in\mathds{R}$.

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