Infinitely many solutions for three quasilinear Laplacian systems on weighted graphs

We investigate a generalized poly-Laplacian system with a parameter on weighted finite graph, a generalized poly-Laplacian system with a parameter and Dirichlet boundary value on weighted locally finite graphs, and a $(p,q)$-Laplacian system with a parameter on weighted locally finite graphs. We utilize a critical points theorem built by Bonanno and Bisci [Bonanno, Bisci, and Regan, Math. Comput. Model. 2010, 52(1-2): 152-160], which is an abstract critical points theorem without compactness condition, to obtain that these three systems have infinitely many nontrivial solutions with unbounded norm when the parameters locate some well-determined range.


Introduction
Assume that G = (V , E) is a graph, where V is the vertex set and E is the edge set.G is usually known as a finite graph when V and E have finite elements, and G is usually known as a locally finite graph when for any x ∈ V , there exist finite y ∈ V satisfying xy ∈ E, where xy represents an edge linking x and y.The weight on any given edge xy ∈ E is denoted by ω xy , which is supposed to satisfy ω xy > 0 and ω xy = ω yx .Moreover, we set deg(x) = y∼x ω xy for any fixed x ∈ V .Here, we use y ∼ x to represent those y linked to x. d(x, y) represents the distance between any two points x, y ∈ V , which is defined by the minimal number of edges linking x to y. Suppose that is a subset in V .If there exists a positive constant D such that d(x, y) ≤ D for all x, y ∈ , then is known as a bounded domain in V .Set which is known as the boundary of .The interior of is represented by • = \∂ , which obviously satisfies • = .
Thereinafter, μ : V → R + is supposed to be a finite measure.Set D w,y u(x) := 1 √ 2 u(x)u(y) w xy μ(x) , (1.1) which is the directional derivative of u : V → R, and then the gradient of u is defined as ∇u(x) := D w,y u(x) y∈V (1.2) that is a vector and is indexed by y.
Then it is obvious that which represents the length of ∇u.Furthermore, the length of m-order gradient of u is represented by |∇ m u| that is defined by ifm is an even number. (1.6) Here, we define ∇ (1.9) For any r ∈ R with r ≥ 1, set equipped by the norm . (1.10) For any u : V → R, according to the distributional sense, we write l as where v ∈ C c (V ) and C c (V ) is the set of all real functions with compact support.Furthermore, a more general operator £ m,l could be defined as w h e nm is an even number, ( for any φ ∈ C c (V ), where l ∈ R with l > 1 and m ∈ N. £ m,p is known as the poly-Laplacian of u as m = 2, and £ m,l degenerates to the l-Laplacian operator as m = 1.Those above concepts and more related details refer to [6] and [10].
In this paper, we focus on the existence of infinitely many solutions for the following generalized poly-Laplacian system on finite graph G = (V , E): where Moreover, if G = (V , E) is a locally finite graph, we focus on the existence of infinitely many solutions for the following generalized poly-Laplacian system with Dirichlet boundary condition: Finally, we are also concerned with the existence of infinitely many solutions for the following (p, q)-Laplacian system on locally finite graph G = (V , E): wherep andq are defined by (1.8) with l = p, q, p ≥ 2 and q ≥ 2, F : The existence and asymptotic properties of nontrivial solutions for quasilinear elliptic equations have been studied extensively on Euclidean domain (for example, [13,14,16,24]).With the development of machine learning, data analysis, social network, image processing and traffic network, the analysis on graphs has attracted some attentions [1-3, 7, 20, 21].In particular, recently, in [10] and [11], Grigor'yan, Lin, and Yang studied several nonlinear elliptic equations on graphs and first established the Sobolev spaces and the variational framework on graphs.Subsequently, there have been some works on p-Laplacian equations and more general poly-Laplacian equations on graphs.For example, in [15], Pinamonti and Stefani studied some semi-linear equations with the poly-Laplacian operator on locally finite graphs.They established some existence results of weak solutions via a variational method by using the continuity properties of the energy functionals.In [19], Shao studied a nonlinear p-Laplacian equation on a locally finite graph.Some existence results of positive solutions and positive ground state solutions are established by exploiting the mountain pass theorem and the Nehari manifold.For more related results, also refer to, for example, [8,9,12,17], and [18].
In addition to the case of single equations, recently, the study of systems on graphs has also yielded some results.For example, in [25], Zhang et al. considered system (1.13) with λ = 1.They supposed that F takes on the super-(p, q) growth and then established the existence result of a nontrivial solution by exploiting the mountain pass theorem.They also established a multiplicity result by utilizing the symmetric mountain pass theorem.In [23], Yu et al. considered (1.14) and system (1.15) with p = q, λ = 1, and F(x, u) = -K(x, u) + W (x, u) for all x ∈ V .By utilizing the mountain pass theorem, they achieved that (1.14) has a nontrivial solution.In [22], Yang and Zhang investigated (1.15) with perturbations and two parameters λ 1 and λ 2 .Under the assumptions that the nonlinearity satisfies a sub-(p, q) conditions, they achieved that system has at least one nontrivial solution by Ekeland's variational principle.When the nonlinearity equipped the super-(p, q) conditions, they established that system has at least one nontrivial solution with positive energy and one nontrivial solution with negative energy by exploiting mountain pass theorem and Ekeland's variational principle.In [17], when h 1 (x) = λa +1 and h 2 (x) = λb + 1, Shao studied (1.15) with p = q.By the Nehari manifold method and some analytical techniques, under some suitable assumptions on the potentials and nonlinear terms, they proved that system possesses a ground state solution (u λ , v λ ) when the parameter λ is large enough.
Our investigation is mainly motivated by the above mentioned works and [4,5].In [4], Bonanno and Bisci established the existence result of a sequence {u n } of critical points for the functional f λ :=λ with λ ∈ R, and got a well-determined interval of the parameter λ.In [5], Bonanno and Bisci obtained that a class of quasilinear elliptic system in the Euclidean framework possesses infinitely many weak solutions by the abstract theorem established in [4].In the present paper, we also apply the critical points theorem developed by Bonanno and Bisci [4] to system (1.13), (1.14), and (1.15), and we obtain that these systems have infinitely many nontrivial solutions with unbounded norm when the parameters λ locate some well-determined ranges.To the best of our knowledge, there seemed to be no works to investigate the existence of infinitely many solutions for equations or systems on finite graph or locally finite graph.Our works are a preliminary attempt in this field.

Preliminaries
In this section, we recall some basic knowledge on the Sobolev space on graph.For more details, refer to [10,22,25].We also recall an abstract critical point theorem built in [4], which is exploited to prove our main results.
Suppose that G = (V , E) is a finite graph.For any fixed m ∈ N and any fixed l ∈ R with l > 1, set where h(x) > 0 for all x ∈ V .W m,l (V ) is a Banach space with finite dimension.Suppose that G = (V , E) is a locally finite graph and is a bounded domain in V .For any fixed l ∈ R with l > 1 and any fixed m ∈ N, set ) is a finite dimensional Banach space since is a finite set.On W m,l 0 ( ), one can also equip the following norm: , where h : V → R and there exists a positive constant h 0 such that h(x) ≥ h 0 .Set the space Lemma 2.1 ([10, 25]) Suppose that G = (V , E) is a finite graph.For any ψ ∈ W m,l (V ), there exists where In particular, there exists a positive constant C(m, l, ), which just depends on m, l, and satisfying for all 1 ≤ θ ≤ +∞ and all u ∈ W m,l 0 ( ), where u ,∞ = max x∈ |u(x)| and μ min, = min x∈ μ(x).Moreover, W m,l 0 ( ) is pre-compact, that is, if {u n } is bounded in W m,l 0 ( ), then up to a subsequence, there exists some u ∈ W m,l 0 ( ) such that u n → u in W m,l 0 ( ).
(b) if δ < +∞, for each λ ∈ (0, 1 δ ), the following alternative holds: either (b 1 ) there exists a global minimum of that is a local minimum of I λ , or (b 2 ) there exists a sequence of pairwise distinct critical points (local minima) of I λ that weakly converges to a global minimum of .

Result and proofs for system (1.13)
In this section, we investigate the generalized poly-Laplacian system (1.13) and obtain the following result. Let Theorem 3.1 Suppose that G = (V , E) is a finite graph and the following conditions hold:

13) possesses an unbounded sequence of solutions.
To prove Theorem 3.1, we work in the space Consider the functional I λ,V : W V → R as Then, under the assumptions of Theorem 3.1, I λ,V ∈ C 1 (W V , R) and for all (φ 1 , φ 2 ) ∈ W V .Furthermore, by the arbitrariness of φ 1 and φ 2 , it can be achieved that Therefore, seeking the solutions for system (1.13) is equivalent to seeking the critical points of I λ,V on W V (see [25] for example).
To apply Lemma 2.4, we shall exploit the functionals V : W V → R and V : W V → R, which are set by and Proof Let {c n } be a real sequence satisfying lim n→∞ c n = +∞ and Next, we claim that there exists n 0 ∈ N such that |u(x)| + |v(x)| ≤ c n for all n ≥ n 0 , all x ∈ V , and all (u, v) ∈ W V with V (u, v) ≤ r n .We prove the claim through the following three cases.Without loss of generality, we let δ = q.
Note that qp ≤ 0. Then the above inequality implies that , by (3.2) and (3.4), we have Hence, an easy computation implies that Thus, based on the three cases, we conclude that for all (u, v) Hence, (F 2 ) implies that This finishes the proof of the lemma.

Lemma 3.2 For any fixed
Proof Assume that {ξ n } and {η n } are two positive real sequences satisfying For each n ∈ N, we define Then, for every n ∈ N, we have where V is defined by (3.1).If B V < +∞, choosing λ ∈ ( V λB , 1), by (3.5), there exists n λ > 0 such that Then, combining with (3.6), we get Then Then, combining with (3.6), we get Noticing the choice of M λ , we also have Thus, we finish the proof of the lemma.

Lemma 3.3 V is sequentially weakly lower semi-continuous.
Proof The proof is easily finished by exploiting the weak lower semi-continuity of the norm.

Lemma 3.4 V is sequentially weakly upper semi-continuous.
Proof Assume that (u n , v n ) (u 0 , v 0 ) in W V .Note that W V is of finite dimension.Then (u n , v n ) → (u 0 , v 0 ) in W V .By (F 0 ) and the fact that V is a finite set, it is easy to obtain that Hence, V is sequentially weakly upper semi-continuous in W V .
Proof of Theorem 3.1 It is easy to see that V : W V → R is coercive.Lemma 3.1-Lemma 3.4 imply that all of conditions in Lemma 2.4 are satisfied.Hence, Lemma 2.4 (a) implies that for each (λ 1,V λ 2,V ), the functional

Result and proofs for system (1.14)
In this section, we investigate the generalized poly-Laplacian system (1.14) and obtain the following result.Let where C(m 1 , p, ) and C(m 2 , q, ) are defined in Lemma 2.2.
The proofs of Theorem 4.1 are the essentially same as Theorem 3.1 with some slight modifications.To prove Theorem 4.1, we work in the space Then, under the assumptions of Theorem 4.1, I λ, ∈ C 1 (W 0 , R) and for all (φ 1 , φ 2 ) ∈ W 0 .Furthermore, by the arbitrariness of φ 1 and φ 2 , it can be achieved that system (1.14) holds.Therefore, seeking the solutions for system (1.14) is equivalent to seeking the critical points of I λ,V on W 0 .To apply Lemma 2.4, we will use the functionals : W 0 → R and : Then I λ, (u, v) = λ and for every r > inf W 0 , define Lemma 4.1 Assume that (F 2 ) holds.Then γ := lim inf r→+∞ ϕ (r) < +∞.
Proof The proof is the same as that of Theorem 3.1 with substituting , K , A , B , u ∞, , and v ∞, for V , K V , A V , B V , u ∞,V , and v ∞,V , respectively.We omit the details.

Lemma 4.2 For any fixed
Proof Suppose that {ξ n } and {η n } are two positive real sequences such that lim n→∞ |ξ n | + |η n | = +∞ and For each n ∈ N, we define , there exists n λ such that Thus, By the choice of M λ , we also have Thus, we finish the proof of this lemma.

Lemma 4.3 sequentially weakly lower semi-continuous.
Proof The proof is easily completed by using the weak lower semi-continuity of the norm.
Lemma 4.4 is sequentially weakly upper semi-continuous.
Proof The proof is the same as that of Lemma 3.4 with replacing W with W 0 and V with .
Proof of Theorem 4.1 It is obvious that : W 0 → R is coercive.Lemma 4.1-Lemma 4.4 imply that all of conditions in Lemma 2.4 are satisfied.Hence, Lemma 2.4(a) implies that for each (λ 1, , λ 2, ), I λ, has a sequence {(u n , v n )} of critical points that are solutions of system (1.14) 5 Result and proofs for system (1.15) In this section, we investigate the (p, q)-Laplacian system (1.15).We first make the following assumptions: (M 1 ) There exists μ 0 > 0 such that μ(x) ≥ μ 0 for all x ∈ V ; (M 2 ) There exists (H 1 ) There exists a constant h 0 > 0 such that h i (x) ≥ h 0 > 0 for all x ∈ V , i = 1, 2; Let Theorem 5.1 Suppose that G = (V , E) is a locally finite graph, and (M 1 ), (M 2 ), (H 1 ) and the following conditions hold: ( F0 ) F(x, s, t) is continuously differentiable in (s, t) ∈ R 2 for all x ∈ V , and there exist a function a ∈ C(R + , R + ) and a function b for all x ∈ V and all (s, t) ∈ R 2 ; ( F1 ) V F(x, 0, 0) dμ = 0; ( F2 ) where δ = min{p, q}.

We work in the space
(V ) and then (W , • ) is a Banach space that is infinite dimensional.We consider the functional I λ : W → R as Then, by Appendix A.2 in [22], under the assumptions of Theorem 5.1, I λ ∈ C 1 (W , R), and for all (φ 1 , φ 2 ) ∈ W . Furthermore, by the arbitrariness of φ 1 and φ 2 , it can be achieved that system (1.15) holds.Therefore, seeking the solutions for system (1.15) is equivalent to seeking the critical points of I λ on W . Define : W → R and : Then I λ (u, v) =λ .For every r > inf , set Lemma 5.1 Assume that ( F2 ) holds.Then γ := lim inf r→+∞ ϕ(r) < +∞.
Proof The is essentially the same as that of Theorem 3.1 with substituting the locally finite graph V , K , A, B, u ∞ , and v ∞ for the finite graph V , K V , A V , B V , u ∞,V , and v ∞,V , respectively.We omit the details.
Proof By ( F2 ), we can assume that {ξ n } and {η n } are two positive real sequences satisfying (5.4) For each n ∈ N, define where x 0 is given in assumption (M 2 ).Then a simple calculation implies that and similarly, where M 1 (x 0 ) and M 2 (x 0 ) are given in assumption (M 2 ).Then {(u n , v n )} ⊂ W and for every n ∈ N, we have where is given in (5.1).If B < +∞, choosing ˜ λ ∈ ( λB , 1), by (5.4), there exists n ˜ λ such that Thus, combining with (5.5), we have If B = +∞, let us consider Mλ > λ .By (5.4), there exists n Mλ such that Combining the choice of Mλ , in this case, we also have Thus we complete the proof of this lemma.
Lemma 5.3 is sequentially weakly lower semi-continuous.
The proofs of Theorem 6.1 are almost the same as those of Theorem 3.1 and even simpler because there is no couple term.Here, we just present the proof that γV := lim inf r→+∞ φV (r) < +∞, which is related to the range of the parameter of λ and also show that the proof for single equation is indeed simpler, where and Ĩλ,V = ˜ Vλ ˜ V is the corresponding variational functional of (6.1).In fact, let {c n } be a real sequence satisfying lim n→∞ c n = +∞ and Then, for each λ ∈ ( λ1, , λ2, ) with λ1, = 1 B and λ2, = 1 p K Ã , where K = C p (m,p, ) μ min, , equation (6.2) possesses an unbounded sequence of solutions.
By using similar arguments as those of Theorem 5.1, we can also obtain similar results for the following scalar equation on locally finite graph G = (V , E): where h : V → R, p ≥ 2, λ > 0, and f : V × R → R. We make the following assumptions: (h) There exists a constant h 0 > 0 such that h(x) ≥ h 0 > 0 for all x ∈ V ; (M) There exists x 0 ∈ V such that M(x 0 ) ≤ M(x) for all x ∈ V , where