Positive solutions of p-th Yamabe type equations on graphs

Abstract

Let G = (V, E) be a finite connected weighted graph, and assume 1 ≤ α ≤ p ≤ q. In this paper, we consider the following p-th Yamabe type equationon G, where ∆ p is the p-th discrete graph Laplacian, h ≤ 0 and f > 0 are real functions defined on all vertices of G. Instead of the approach in [12], we adopt a new approach, and prove that the above equation always has a positive solution u > 0 for some constant λ ∈ R. In particular, when q = p our result generalizes the main theorem in [12] from the case of α ≥ p > 1 to the case of 1 ≤ α ≤ p. It's interesting that our new approach can also work in the case of α ≥ p > 1. critical points of the Hilbert-Einstein function S on the space of Riemannian metrics on M restricted to [g], the conformal class of g, S(g) = M s g dµ g V ol(M, g) 2/pm .

A class of semilinear elliptic equations on lattice graphs

“…Recently, people began to consider semilinear equations on discrete spaces. For example, a class of semilinear equations with the nonlinearity of power type, including the well known Yamabe type equations, have been studied on graphs, see [8,16,17,20,[24][25][26]28,[45][46][47]. A class of semilinear equations with the exponential nonlinearity, so-called Kazdan-Warner equations and the Liouville equations, also have been studied in these papers [15,18,19,23,32,44] on graphs.…”

“…However, we remove the constraint that q < 2 * (subcritical) since we have the embedding l s into l t for s < t in the discrete setting. (ii) In the discrete setting, Ge, Zhang and Lin [16,45] proved the p-th Yamabe equation has a positive solution for one pair of coefficients on finite graphs. While we can prove that for any ã > 0, there are infinitely many b > 0 such that the equation ( 4) has at least a positive solution.…”

A 𝑝-th Yamabe equation on graph

“…Under some additional assumptions on the graphs and functions, Ge-Jiang [10], Keller-Schwarz [17] get some results with totally different techniques and assumptions. In [12], Grigor'yan-Lin-Yang studied the Yamabe type equation −∆u − αu = |u| p−2 u on a finite domain Ω of an infinite graph, with u = 0 outside Ω. Ge [7] studied the p-th Yamabe equation ∆ p u + hu p−1 = λf u α−1 on a finite graph under the assumption α ≥ p > 1, which was generalized to α < p by Zhang-Lin [26]. Ge-Jiang [9] further generalized Ge's results to get a global positive solution on an infinite graph under suitable assumptions.…”

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

“…Recently, the investigations of discrete weighted Laplacians and various equations on graphs have attracted much attention (cf. [1,2,3,4,5,6,7,8]). Grigor'yan, Lin and Yang [3] first studied a Yamabe type equation on a finite graph G as follows − ∆u + hu = |u| α−2 u, α > 2 (1.1) where ∆ is a usual discrete graph Laplacian, and h is a positive function defined on the vertices of G. They show that the above equation (1.1) always has a positive solution.…”