EXISTENCE OF GLOBAL SOLUTIONS TO SOME NONLINEAR EQUATIONS ON LOCALLY FINITE GRAPHS

Abstract

. Let G = ( V,E ) be a connected locally finite and weighted graph, ∆ p be the p -th graph Laplacian. Consider the p -th nonlinear equation − ∆ p u + h | u | p − 2 u = f ( x,u ) on G , where p > 2, h,f satisfy certain assumptions. Grigor’yan-Lin-Yang [24] proved the existence of the solution to the above nonlinear equation in a bounded domain Ω ⊂ V . In this paper, we show that there exists a strictly positive solution on the infinite set V to the above nonlinear equation by modifying some conditions in [24]. To the m -order differential operator L m,p , we also prove the existence of the nontrivial solution to the analogous nonlinear equation.