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

Let G = (V;E) be a connected locally finite and weighted graph, Delta(p) be the p-th graph Laplacian. Consider the p-th nonlinear equation -Delta(p)u + h vertical bar u vertical bar- 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 Omega subset of 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,L-p, we also prove the existence of the nontrivial solution to the analogous nonlinear equation.