有限图上高阶Yamabe型方程的非平凡解

Nontrivial Solution of High Order Yamabe-Type Equation on Finite Graph

该文研究了以下高阶Yamabe型方程Lm,pu−g|u|^p−2u=λf|u|α−2u在有限图上的非平凡正解的存在性,其中Lm,p是一个2m阶差分算子,它是一种p次(−Δ)^m算子更一般化,α≥p≥2,g>0和f>0是定义在G的所有顶点上的实函数,m≥1是一个整数.

In this paper,we study the existence of nontrivial positive solution of the following high order Yamabe-type equation Lm,pu−g|u|^p−2u=λf|u|α−2u on a finite graph G,where Lm,p is a 2m-order difference operator which is a kind of p-th(−Δ)^m operator,α≥p≥2,g>0 and f>0 are real functions defined on all vertices of G,m≥1 is an integer.We show that the above equation always has a nontrivial solution u≥0 for some constantλ∈R.