Existence of positive solutions to some nonlinear equations on locally finite graphs

Let $G=(V,E)$ be a locally finite graph, whose measure $\mu(x)$ have positive lower bound, and $\Delta$ be the usual graph Laplacian. Applying the mountain-pass theorem due to Ambrosetti-Rabinowitz, we establish existence results for some nonlinear equations, namely $\Delta u+hu=f(x,u)$, $x\in V$. In particular, we prove that if $h$ and $f$ satisfy certain assumptions, then the above mentioned equation has strictly positive solutions. Also, we consider existence of positive solutions of the perturbed equation $\Delta u+hu=f(x,u)+\epsilon g$. Similar problems have been extensively studied on the Euclidean space as well as on Riemannian manifolds.