The pth Kazdan–Warner equation on graphs

Abstract

Let [Formula: see text] be a connected finite graph and [Formula: see text] be the set of functions defined on [Formula: see text]. Let [Formula: see text] be the discrete [Formula: see text]-Laplacian on [Formula: see text] with [Formula: see text] and [Formula: see text], where [Formula: see text] is positive everywhere. Consider the operator [Formula: see text]. We prove that [Formula: see text] is one-to-one, onto and preserves order. So it implies that there exists a unique solution to the equation [Formula: see text] for any given [Formula: see text]. We also prove that the equation [Formula: see text] has a solution which is unique up to a constant, where [Formula: see text] is the average of [Formula: see text]. With the help of these results, we finally give various conditions such that the [Formula: see text]th Kazdan–Warner equation [Formula: see text] has a solution on [Formula: see text] for given [Formula: see text] and [Formula: see text]. Thus we generalize Grigor’yan, Lin and Yang’s work Kazdan–Warner equation on graph, Calc. Var. Partial Differential Equations 55(4) (2016) Paper No. 92, 13 pp. for [Formula: see text] to any [Formula: see text].