Yamabe equations on infinite graphs

We concern in this paper the graph Yamabe equation Δu+hu=g|u|p−2u, with known functions h and g on an infinite graph, the prototype of which comes from the smooth Yamabe equation on an open manifold. We prove the existence of a solution to the graph Yamabe equation under the assumption that (1) the graph Laplacian Δ is a bounded operator and (2) g is bounded and h is large at infinity.