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

Zhang, Xiaoxiao; Lin, Aijin

doi:10.1090/proc/14362

Cited by 31 publications

(10 citation statements)

References 11 publications

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“…Step 4. There exists a subsequence of (u k ), which is still denoted by (u k ), and a function u * : V → R such that (u k ) converges to u * locally uniformly in V. Moreover, u * is a solution of the equation (16).…”

Section: The Case Gmentioning

confidence: 99%

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Calculus of variations on locally finite graphs

Lin¹,

Yang²

2021

Preprint

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Let G = (V, E) be a locally finite graph. Firstly, using calculus of variations, including a direct method of variation and the mountain-pass theory, we get sequences of solutions to several local equations on G (the Schrödinger equation, the mean field equation, and the Yamabe equation). Secondly, we derive uniform estimates for those local solution sequences. Finally, we obtain global solutions by extracting convergent sequence of solutions. Our method can be described as a variational method from local to global.

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Section: The Case Gmentioning

confidence: 99%

“…The Kazdan-Warner equation was extended by Keller-Schwarz [11] to canonically compactifiable graphs, and by Ge-Jiang [5] to certain infinite graph. For other related works, we refer the reader to [10,12,13,9,14,16] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Calculus of variations on locally finite graphs

Lin¹,

Yang²

2021

Preprint

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“…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.…”

Section: Introductionmentioning

confidence: 99%

A class of semilinear elliptic equations on lattice graphs

Hua¹,

Li²,

Wang³

2022

Preprint

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In this paper, we study the semilinear elliptic equation of the formon lattice graphs Z N , where N ≥ 2 and 2 ≤ p < q < +∞. By the Brézis-Lieb lemma and concentration compactness principle, we prove the existence of positive solutions to the above equation with constant coefficients ā, b and the decomposition of bounded Palais-Smale sequences for the functional with variable coefficients, which tend to some constants ā, b at infinity, respectively.

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“…on an infinite graph. The main result in [13] is as follows We will solve this problem in our another paper [21].…”

Section: Introductionmentioning

confidence: 99%

Positive solutions of p-th Yamabe type equations on graphs

Zhang¹,

Lin²

2018

Front. Math. China

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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 .

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

Cited by 31 publications

References 11 publications

Calculus of variations on locally finite graphs

Calculus of variations on locally finite graphs

A class of semilinear elliptic equations on lattice graphs

Positive solutions of p-th Yamabe type equations on graphs

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