Existence of positive solutions to some nonlinear equations on locally finite graphs
نویسندگان
چکیده
Let G = (V, E) be a locally finite graph, whose measure μ(x) have positive lower bound, and ∆ be the usual graph Laplacian. Applying the mountain-pass theorem due to Ambrosetti-Rabinowitz, we establish existence results for some nonlinear equations, namely ∆u + hu = f (x, u), x ∈ 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 ∆u + hu = f (x, u) + g. Similar problems have been extensively studied on the Euclidean space as well as on Riemannian manifolds.
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