The $1$-Yamabe equation on graph

Abstract

We study the following $1$-Yamabe equation on a connected finite graph $$Δ_1u+g\mathrm{Sgn}(u)=h|u|^{α-1}\mathrm{Sgn}(u),$$ where $Δ_1$ is the discrete $1$-Laplacian, $α>1$ and $g, h>0$ are known. We show that the above $1$-Yamabe equation always has a nontrivial solution $u\geq0$, $u\neq0$.

10 pages. All comments are welcome