SOME GLOBAL EXISTENCE RESULTS ON LOCALLY FINITE GRAPHS

Abstract Let $G=(V, E)$ be a locally finite graph with the vertex set V and the edge set E , where both V and E are infinite sets. By dividing the graph G into a sequence of finite subgraphs, the existence of a sequence of local solutions to several equations involving the p -Laplacian and the poly-Laplacian systems is confirmed on each subgraph, and the global existence for each equation on graph G is derived by the convergence of these local solutions. Such results extend the recent work of Grigor’yan, Lin and Yang [ J. Differential Equations , 261 (2016), 4924–4943; Rev. Mat. Complut. , 35 (2022), 791–813]. The method in this paper also provides an idea for investigating similar problems on infinite graphs.