The $1$-Yamabe equation on graph

Abstract

We study the following $1$-Yamabe equation on a connected finite graph $${\Delta }_{1}u+g\mathrm{Sgn}(u)=h|u{|}^{\alpha -1}\mathrm{Sgn}(u),$$ where ${\Delta }_1$ is the discrete $1$-Laplacian, $\alpha >1$ and $g,h>0$ are known. We show that the above $1$-Yamabe equation always has a nontrivial solution $u\ge 0$, $u\ne 0$.

10 pages. All comments are welcome

References in corpus (3)

• Kazdan-Warner equation on infinite graphs
• p-th Kazdan-Warner equation on graph
• The Kazdan-Warner equation on canonically compactifiable graphs