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Let G = (V,E) be a connected infinite and locally finite weighted graph, ∆p be the p-th discrete graph Laplacian. In this paper, we consider the p-th Yamabe type equation −∆pu+ h|u| u = gu on G, where h and g are known, 2 < α ≤ p. The prototype of this equation comes from the smooth Yamabe equation on an open manifold. We prove that the above equation has at least one positive solution on G.

Positive solutions of $p$-th Yamabe type equations on infinite graphs