Multiple Solutions of a p-th Yamabe Equation on Graph

Abstract

Let G=V,E be a connected finite graph and Δp be the p-Laplacian on G with p>1. We consider a perturbed p-th Yamabe equation −Δpu−λup−2u=huα−2u+εf, where h,f:V⟶ℝ are functions with h,f>0; 1<p<α;λ and ε are two positive constants. Using the variational method, we prove that there exists some positive constant ϵ1 such that for all ϵ∈0,ϵ1, the above equation has two distinct solutions.