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