Existence and multiplicity of nontrivial solutions for poly-Laplacian systems on finite graphs

In this paper, we investigate the existence and multiplicity of nontrivial solutions for poly-Laplacian system on a finite graph G=(V,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$G=(V, E)$\end{document}, which is a generalization of the Yamabe equation on a finite graph. When the nonlinear term F satisfies the super-(p,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p, q)$\end{document}-linear growth condition, by using the mountain pass theorem we obtain that the system has at least one nontrivial solution, and by using the symmetric mountain pass theorem, we obtain that the system has at least dim W nontrivial solutions, where W is the working space of the poly-Laplacian system. We also obtain the corresponding result for the poly-Laplacian equation. In some sense, our results improve some results in (Grigor’yan et al. in J. Differ. Equ. 261(9):4924–4943, 2016).


Introduction
In this paper, we mainly consider the following high-order Yamabe-type coupled system, which is called the poly-Laplacian system: where V is a finite graph, m i ≥ 2, i = 1, 2, p, q > 1 are integers, h i : V → R + , i = 1, 2, F : V × R × R → R, and £ m,p is defined as follows: for any function φ : V → R, When p = 2, £ m,p = (-) m u is called the poly-Laplacian operator of u, and when m = 1, £ m,p =p u. A detailed definition is given in Sect. 2; see also [1].
When m 1 = m 2 = m, p = q, and u = v, system (1.1) becomes the scalar equation where f (x, u) = F u (x, u) for x ∈ V , and can be seen as a generalization of the following Yamabe equation on a finite graph: In recent years, some scholars are devoted to studying the Yamabe equation on finite and infinite graphs. We refer the readers to [1][2][3][4][5][6][7]. Ge [2] studied the following Yamabe-type equations with p-Laplacian operator on finite graphs: where 1 < m -1 ≤ α, f > 0, h > 0, and p is defined by where ω xy is the weight of the edge connecting x and y. When the nonlinear term f > 0, m = p -1, and λ ∈ R, Ge established the existence of a positive solution. When 1 ≤ α ≤ p ≤ q, h ≤ 0, and f > 0, Zhang [3], extended the case of m = p -1 in (1.5) to m = q -1 and proved the existence of a positive solution. Ge and Jiang [5] and Zhang and Lin [6] extended the existence results of solutions on finite graphs to infinite graphs for p = 2 and p > 2 and obtained the existence of one positive solution. Han and Shao [4] investigated the nonlinear p-Laplacian equation where p ≥ 2, where the definition of the p-Laplacian operator p is different: Under appropriate conditions on the nonlinear terms f (x, u) and a(x), the author obtained the existence of a positive solution for equation (1.6) via the mountain pass theorem. Pinamonti and Stefani [7] studied the following equation with the (m, p)-Laplacian operator on locally finite weighted graphs: where • and ∂ are the interior and boundary of , respectively. They established the existence of at least one nontrivial solution when 0 < λ < for some > 0 via the varia-tional method. Besides, they also investigated the following Yamabe-type equations" where f ∈ L 1 ( ), h ∈ L 1 (∂ ), and g : × R → R is a function such that g(x, 0) = 0 and t → g(x, t) is nondecreasing for all x ∈ . They obtained the uniqueness of weak solutions. The research of this paper is mainly inspired by a recent work due to Grigor'yan, Lin, and Yang [1], who investigated the Yamabe equation and its generalization, that is, poly-Laplacian equation on locally finite and finite graphs. To be specific, in [1], for equation (1.3) on a finite graph V , they assumed that h(x) > 0 for all x ∈ V and F satisfies the following conditions: where λ mp is the first eigenvalue of the operator £ m,p , and They obtained the existence of a nontrivial solution via the mountain pass theorem.
In this paper, we would like to generalize and improve the above result in [1]. We use the mountain pass theorem to study the existence of a nontrivial solution and use the symmetric mountain pass theorem to study the multiplicity of nontrivial solutions for system (1.1) on a finite graph, where the nonlinear term F satisfies the super-(p, q)-linear growth condition. Our work is also inspired by Luo and Zhang [8], who considered the following nonlinear p-Laplacian difference system: where p ≥ 2, φ p (s) = |s| p-2 s, u(n) = u(n + 1)u(n), F(n, x) is continuously differentiable in x for all n ∈ {1, . . . , M}, and M > 1 is a positive integer. By the linking theorem in [9] they obtained that the system has at least one nonconstant periodic solution when F satisfies super-p-linear growth condition. Next, we state our main results.
Theorem 1.1 Assume that F satisfies the following conditions: Then system (1.1) has at least one nontrivial solution. Theorem 1.2 Assume that (F 1 )-(F 4 ) and the following condition hold: From Theorems 1.1 and 1.2 we easily obtain the following results corresponding to (1.3).

Theorem 1.3 Assume that F satisfies the following conditions:
3) has at least one nontrivial solution.
Theorem 1.4 Assume that (F 1 )-(F 4 ) and the following condition hold: Remark 1.4 In some sense, (F 1 )-(F 4 ) can be seen as a generalization of the assumptions in [8], where the difference equation (1.7) is studied, defined on the set Z of integers. However, in this paper, we study the high-order Yamabe-type coupled system involving the poly-Laplacian on a finite graph. Hence we generalize those conditions in [8] from m = 1 to m ≥ 2 and from n ∈ Z to x ∈ V , which is a finite graph. Moreover, we also present the multiplicity results, that is, Theorems 1.2 and 1.4, which are not considered in [1].

Preliminaries
In this section, we state some useful properties of poly-Laplacian and Sobolev spaces on graphs. For details, we refer to [1].
Let G = (V , E) be a finite graph with vertex set V and edge set E. For any edge xy ∈ E with two vertexes of x, y ∈ V , assume that its weight ω xy > 0 and ω xy = ω yx . For any x ∈ V , its degree is defined as deg( The corresponding gradient form is Write (ψ) = (ψ, ψ). The length of the gradient is defined by and |V | = x∈V μ(x). When p ≥ 2, we define the p-Laplacian operator by p ψ by In the distributional sense, p ψ can be written as follows. For any φ ∈ C c (V ), where C c (V ) is the set of all real functions with compact support. It is easy to see that £ m,p defined by (1.2) is a generalization of p ψ.
Define the space Let X be a Banach space, and let ϕ ∈ C 1 (X, R). We say that the functional ϕ satisfies the Palais-Smale (PS) condition if {u n } has a convergent subsequence in X whenever ϕ(u n ) is bounded and ϕ (u n ) → 0. We call that ϕ satisfies the Cerami (C) condition if {u n } has a convergent subsequence in X whenever ϕ(u n ) is bounded and ϕ (u n ) × (1 + u n ) → 0. Lemma 2.1 (Mountain pass theorem [10]) Let X be a real Banach space, and let ϕ ∈ C 1 (X, R), ϕ(0) = 0 satisfy the (PS)-condition. Suppose that ϕ satisfies the following conditions:
Remark 2.2 If X is finite-dimensional, the result of Lemma 2.2 can also be obtained with the conclusion that ϕ possesses at least dim Z critical values (see [10], Remark 9.36).

Lemma 2.4 Let G = (V , E) be a finite graph. Let m be any positive integer, and let q > 1.
Then W m,p (V ) → L q (V ) for all 1 ≤ q ≤ +∞. In particular, if 1 < q < +∞, then for all ψ ∈ W m,p (V ),

Proofs of main results
Note that the space Then ϕ ∈ C 1 (W , R), and By the arbitrariness of φ 1 and φ 2 we conclude that Thus the problem of finding the solutions of system (1.1) is reduced to finding the critical points of the functional ϕ on W .

Lemma 3.1
Assume that (F 4 ) holds. Then the functional ϕ satisfies condition (C), that is, {(u k , v k )} has a convergent subsequence in W whenever ϕ(u k , v k ) is bounded and Then there exists a positive constant L such that for every k ∈ N. By (F 4 ),there are constants C 1 > 0 and δ 1 > 0 such that for all (t, s) ∈ R 2 and x ∈ V , where Then for all large k, we have When max{p, q} = p, Therefore v k W m 2 ,q (V ) , u k L γ 1 (V ) , and v k L γ 2 (V ) are bounded. Since (W , · ) is a finitedimensional space, there exist positive constants D 1 and D 2 such that Hence ϕ satisfies the (C)-condition.

Lemma 3.2 There exists a constant
Proof By (F 2 ) there are 0 < C 4 < min{ 1 } and a positive constant for all |(t, s)| ≤ δ 2 . By Lemma 2.4 we have Then by (3.5) and (3.6), for all (u, v) ∈ W with (u, v) = ρ, we have The proof is completed.
Hence the associated point (u * , v * ) ∈ W is a nontrivial weak solution of system (1.1).
Proof Let dim X = m. Then there exist positive constants C 6 (m) and C 7 (m) such that for all (u, v) ∈ X. By (F 3 ) we know that there exist constants β > C 6 (m) p p + C 7 (m) q q and r > 0 such that F(x, t, s) ≥ β |t| p + |s| q for all (t, s) ≥ r and x ∈ V . (3.8) It follows from (F 1 ) and (3.8) that there exists C 8 > 0 such that F(x, t, s) ≥ β |t| p + |s| q -C 8 for all (t, s) ∈ R 2 and x ∈ V . for all (u, v) ∈ X. Note that β > C 6 (m) p p + C 7 (m) q q . So ϕ(u, v) → -∞ as (u, v) → ∞. Thus we complete the proof.