Abstract:

In this paper, we study the existence of nontrivial positive solution of the following high order Yamabe-type equation ${\cal L}_{m, p} u-g|u|^{p-2} u=\lambda f|u|^{\alpha-2} u $ on a finite graph $ G$, where $ {\cal L}_{m, p}$ is a $ 2m$-order difference operator which is a kind of $ p$-th $ (-\Delta)^m$ operator, $ \alpha \geq p \geq 2$, $g>0 $ and $f>0 $ are real functions defined on all vertices of $G $, $ m\ge 1$ is an integer. We show that the above equation always has a nontrivial solution $u\ge 0 $ for some constant λ∈ ${\Bbb R} $.

Key words: Finite graph, High order Yamabe-type equation, Nontrivial positive solution