A p-th Yamabe equations on graph

Abstract

Assume $\alpha \ge p>1$. Consider the following $p$-th Yamabe equation on a connected finite graph $G$: $${\Delta }_p\phi +h{\phi }^{p-1}=\lambda f{\phi }^{\alpha -1},$$ where ${\Delta }_p$ 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 $\phi$ for some constant $λ\in\mathds{R}$.

7 pages