Home Existence of three solutions for two quasilinear Laplacian systems on graphs
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Existence of three solutions for two quasilinear Laplacian systems on graphs

  • Yan Pang and Xingyong Zhang EMAIL logo
Published/Copyright: November 18, 2024
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 57 Issue 1

Abstract

We deal with the existence of three distinct solutions for a poly-Laplacian system with a parameter on finite graphs and a ( p , q ) -Laplacian system with a parameter on locally finite graphs. The main tool is an abstract critical point theorem in [G. Bonanno and S. A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition, Appl. Anal. 89 (2010), no. 1, 1–10]. A key point in this study is that we overcome the difficulty to prove that the Gâteaux derivative of the variational functional for poly-Laplacian operator admits a continuous inverse, which is caused by the special definition of the poly-Laplacian operator on graph and mutual coupling of two variables in system.

Keywords: critical point theorem; ploy-Laplacian system; finite graph; locally finite graph; nontrivial solution
MSC 2010: 35J60; 35J62; 49J35

1 Introduction

Let G = ( V , E ) be a graph with the vertex set V and the edges set E . If both V and E are finite set, then G is called as a finite graph. If for any x ∈ V , there are finite vertexes y ∈ V such that x y ∈ E ( x y denotes an edge connecting x with y ), then G is called as a locally finite graph. For any edge x y ∈ E with two vertexes x , y , let ω x y ( > 0 ) denote its weight and suppose that ω x y = ω y x . For any x ∈ V , let deg ( x ) = ∑ y ∼ x ω x y , where y ∼ x denotes those y connecting x with x y ∈ E . Suppose that μ : V → R + is a finite measure. Define the directional derivative of a function u : V → R by

(1.1) D w , y u ( x ) ≔ 1 2 ( u ( x ) − u ( y ) ) w x y μ ( x ) .

Define the gradient of u as a vector

(1.2) ∇ u ( x ) ≔ ( D w , y u ( x ) ) y ∈ V ,

which is indexed by the vertices y ∈ V . It is easy to obtain that ∇ ( u 1 + u 2 ) = ∇ u 1 + ∇ u 2 and

∇ u ∇ v = ( D w , y u ( x ) ) y ∈ V ( D w , y v ( x ) ) y ∈ V = ∑ y ∼ x D w , y u ( x ) D w , y v ( x ) = 1 2 μ ( x ) ∑ y ∼ x w x y ( u ( y ) − u ( x ) ) ( v ( y ) − v ( x ) ) .

Let

(1.3) Γ ( u , v ) ( x ) = 1 2 μ ( x ) ∑ y ∼ x w x y ( u ( y ) − u ( x ) ) ( v ( y ) − v ( x ) ) .

Then,

(1.4) Γ ( u , v ) = ∇ u ∇ v .

Let Γ ( u ) = Γ ( u , u ) and define the length of the gradient by

(1.5) ∣ ∇ u ∣ ( x ) = Γ ( u ) ( x ) = 1 2 μ ( x ) ∑ y ∼ x w x y ( u ( y ) − u ( x ) ) 2 1 2 .

The Laplacian operator of u : V → R is defined by

(1.6) Δ u ( x ) ≔ 1 μ ( x ) ∑ y ∼ x w x y ( u ( y ) − u ( x ) ) .

Let ∣ ∇ m u ∣ denote the length of m -order gradient of u , which is defined by

(1.7) ∣ ∇ m u ∣ = ∣ ∇ Δ m − 1 2 u ∣ , when  m  is odd, ∣ Δ m 2 u ∣ , when  m  is even,

where ∇ Δ m − 1 2 u is defined as in (1.2) with u replaced by Δ m − 1 2 u , and Δ m 2 u is defined by Δ m 2 u = Δ ( Δ m 2 − 1 u ) which means that u is replaced by Δ m 2 − 1 u in (1.6). By mathematical induction, we can obtain that

(1.8) Δ m 2 ( u 1 + u 2 ) = Δ m 2 u 1 + Δ m 2 u 2 , if m is even.

For any given p > 1 , the p -Laplacian operator is defined by

(1.9) Δ p u ( x ) ≔ 1 2 μ ( x ) ∑ y ∼ x ( ∣ ∇ u ∣ p − 2 ( y ) + ∣ ∇ u ∣ p − 2 ( x ) ) ω x y ( u ( y ) − u ( x ) ) .

It is easy to see that p -Laplacian operator reduces to the Laplacian operator of u if p = 2 .

For any function u : V → R , we denote

(1.10) ∫ V u ( x ) d μ = ∑ x ∈ V μ ( x ) u ( x ) .

For any given real number r ≥ 1 , we define

L r ( V ) = u : V → R ∫ V ∣ u ( x ) ∣ r d μ < ∞

endowed with the norm

(1.11) ‖ u ‖ L r ( V ) = ∫ V ∣ u ( x ) ∣ r d μ 1 r .

In the distributional sense, Δ p u can be written as follows. For any u ∈ C c ( V ) ,

(1.12) ∫ V ( Δ p u ) v d μ = − ∫ V ∣ ∇ u ∣ p − 2 Γ ( u , v ) d μ ,

where C c ( V ) is the set of all real functions with compact support. Furthermore, a more general operator can be introduced, denoted by £ m , p , as follows: for any function ϕ : V → R ,

(1.13) ∫ V ( £ m , p u ) ϕ d μ = ∫ V ∣ ∇ m u ∣ p − 2 Γ ( Δ m − 1 2 u , Δ m − 1 2 ϕ ) d μ , if  m  is odd , ∫ V ∣ ∇ m u ∣ p − 2 Δ m 2 u Δ m 2 ϕ d μ , if  m  is even ,

where m ≥ 1 are integers and p > 1 . The operator £ m , p is called as the poly-Laplacian operator of u if p = 2 , and obviously, the operator £ m , p reduces to the p -Laplacian operator if m = 1 . The results above are taken from [1,2].

In this study, we study the existence of three solutions for the following poly-Laplacian system:

(1.14) £ m 1 , p u + h 1 ( x ) ∣ u ∣ p − 2 u = λ F u ( x , u , v ) , x ∈ V , £ m 2 , q v + h 2 ( x ) ∣ v ∣ q − 2 v = λ F v ( x , u , v ) , x ∈ V ,

where G = ( V , E ) is a finite graph, m i ≥ 1 are integers, h i : V → R , i = 1 , 2 , p , q > 1 , λ > 0 , F : V × R 2 → R , and £ m 1 , p and £ m 2 , q are defined by (1.13).

Moreover, if G = ( V , E ) is a locally finite graph, we consider the existence of three solutions for the following ( p , q ) -Laplacian system:

(1.15) − Δ p u + h 1 ( x ) ∣ u ∣ p − 2 u = λ F u ( x , u , v ) , x ∈ V , − Δ q v + h 2 ( x ) ∣ v ∣ q − 2 v = λ F v ( x , u , v ) , x ∈ V ,

where − Δ p and − Δ q are defined by (1.9) with p ≥ 2 and q ≥ 2 , F : V × R 2 → R , h i : V → R , i = 1 , 2 , and λ > 0 .

In recent years, the study of equations on graphs attracted much attention. We refer readers to [2–8] and references therein. Grigior’yan et al. [2] investigated the following poly-Laplacian equation on graph G = ( V , E ) :

(1.16) £ m , p u + h ( x ) ∣ u ∣ p − 2 u = λ f ( x , u ) in x ∈ V ,

where p > 1 , h : V → R , and f : V × R → R . They considered the case that the graph G = ( V , E ) is a locally finite graph, h ( x ) ≡ 0 and equation (1.16) satisfies the Dirichlet boundary condition, and the case that the graph G = ( V , E ) is a finite graph. They established some existence results of a positive solution for equation (1.16) with λ = 1 by mountain pass theorem. Grigior’yan et al. [3] studied (1.16) with m = 1 and p = 2 , where V is a locally finite graph. They obtained two existence results of positive solutions for equation (1.16) by mountain pass theorem. In [4], when m = 1 and p = 2 , by applying a three critical point theorem from [9], Imbesi et al. established some existence results of at least two solutions for equation (1.16) when the parameter λ locates at some concrete range. Pinamonti and Stefani [5] investigated (1.16) with h ≡ 0 and Dirichlet boundary value condition. They established some existence and uniqueness results. Yu et al. [6] studied system (1.14) with λ = 1 , p = q , and F ( x , u , v ) satisfying asymptotically- p -linear growth at infinity with respect to ( u , v ) . By using the mountain pass theorem, they obtained that system (1.14) has a nontrivial solution and they also presented some corresponding results for equation (1.16) with λ = 1 . Yang and Zhang [7] investigated system (1.15) with perturbations and two parameters λ 1 and λ 2 . When F possesses sub- ( p , q ) growth on ( u , v ) , an existence result of one nontrivial solution was established by Ekeland’s variational principle, and when F possesses super- ( p , q ) growth on ( u , v ) , one solution of positive energy and one solution of negative energy were obtained by mountain pass theorem and Ekeland’s variational principle, respectively. Zhang et al. [10] considered system (1.14) with λ = 1 . They established an existence result and a multiplicity result of nontrivial solutions when F satisfies the super- ( p , q ) growth conditions on ( u , v ) via mountain pass theorem and symmetric mountain pass theorem, respectively.

In the present study, our work are mainly motivated by [10–12]. Bonanno and Marano [11] established an existence result of three critical points for f λ ≔ Φ − λ Ψ with λ ∈ R , and obtained a well-determined large interval of parameters for which f λ possesses at least three critical points under weaker regularity and compactness conditions. Furthermore, by using the three critical points theorem, Bonanno and Bisci [12] obtained that a non-autonomous elliptic Dirichlet problem possesses at least three weak solutions.

Based on the works in [10–12], the motivation of our work is to consider whether the three critical points theorem due to Bonanno and Marano in [11] can be applied to systems (1.14) and (1.15). The main difficulty of such problem is to prove that the Gâteaux derivative of the variational functional Φ admits a continuous inverse. The main reason to cause this difficulty is that the special definition of £ m , p and mutual coupling of u and v in systems (1.14) and (1.15) make proving the uniformly monotone of Φ ′ difficult. To overcome this difficulty, we discuss the case that m is even and the case that m is odd, respectively, and sufficiently use the formulas (1.4) and (1.8), and in order to deal with mutual coupling of u and v , we divide into four cases about the norms of u and v and then, make some careful arguments in Lemma 3.5.

We call that ( u , v ) is a non-trivial solution of system (1.14) (or (1.15)) if ( u , v ) satisfies (1.14) (or (1.15)) and ( u , v ) ≠ ( 0 , 0 ) . Next, we state our results.

(I) For the poly-Laplacian system on finite graph

Theorem 1.1

Let G = ( V , E ) be a finite graph. Assume that the following conditions hold:

  1. h i ( x ) > 0 for all x ∈ V , i = 1 , 2 ;

  2. F ( x , s , t ) is continuously differentiable in ( s , t ) ∈ R 2 for all x ∈ V ;

  3. ∫ V F ( x , 0 , 0 ) d μ = 0 ;

  4. there exist two constants α ∈ [ 0 , p ) , β ∈ [ 0 , q ) and functions f i , g : V → R , i = 1 , 2 , such that

    F ( x , s , t ) ≤ f 1 ( x ) ∣ s ∣ α + f 2 ( x ) ∣ t ∣ β + g ( x )

    for all ( x , s , t ) ∈ V × R × R ;

  5. there are positive constants γ 1 , γ 2 , δ 1 , and δ 2 with δ i > γ i κ i , i = 1 , 2 , such that

    Λ 1 ≔ 1 γ 1 p + γ 2 q max x ∈ V , ∣ s ∣ ≤ ( p γ 1 p + p γ 2 q ) 1 p h 1 , min 1 ⁄ p μ min 1 ⁄ p , ∣ t ∣ ≤ ( q γ 1 p + q γ 2 q ) 1 q h 2 , min 1 ⁄ q μ min 1 ⁄ q F ( x , s , t ) ∣ V ∣ < inf x ∈ V F ( x , δ 1 , δ 2 ) ∣ V ∣ δ 1 p p ∫ V h 1 ( x ) d μ + δ 2 q q ∫ V h 2 ( x ) d μ ≔ Λ 2 ,

    where ∣ V ∣ = ∑ x ∈ V μ ( x ) , h i , min = min x ∈ V h i ( x ) , i = 1 , 2 , μ min = min x ∈ V μ ( x ) , and

    κ 1 = 1 p ∫ V h 1 ( x ) d μ − 1 p , κ 2 = 1 q ∫ V h 2 ( x ) d μ − 1 q .

    Then, for each parameter λ belonging to ( Λ 2 − 1 , Λ 1 − 1 ) , system (1.14) has at least three distinct solutions.

(II) For the (p,q)-Laplacian system on locally finite graph

Theorem 1.2

Let G = ( V , E ) be a locally finite graph. Assume that the following conditions hold:

( M ) there exists a μ 0 > 0 such that μ ( x ) ≥ μ 0 for all x ∈ V ;

( H 1 ) there exists a constant h 0 > 0 such that h i ( x ) ≥ h 0 > 0 for all x ∈ V , i = 1 , 2 ;

( F 0 ) ′ F ( x , s , t ) is continuously differentiable in ( s , t ) ∈ R 2 for all x ∈ V , and there exists a function a ∈ C ( R + , R + ) and a function b : V → R + with b ∈ L 1 ( V ) such that

∣ F s ( x , s , t ) ∣ ≤ a ( ∣ ( s , t ) ∣ ) b ( x ) , ∣ F t ( x , s , t ) ∣ ≤ a ( ∣ ( s , t ) ∣ ) b ( x ) , ∣ F ( x , s , t ) ∣ ≤ a ( ∣ ( s , t ) ∣ ) b ( x ) ,

for all x ∈ V and all ( s , t ) ∈ R 2 ;

( F 1 ) ′ ∫ V F ( x , 0 , 0 ) d μ = 0 and there exists a x 0 ∈ V such that F ( x 0 , 0 , 0 ) = 0 ;

( F 2 ) ′ there exist two constants α ∈ [ 0 , p ) , β ∈ [ 0 , q ) and functions f i , g : V → R , i = 1 , 2 , with f i ∈ L ∞ ( V ) , i = 1 , 2 and g ∈ L 1 ( V ) , such that

F ( x , s , t ) ≤ f 1 ( x ) ∣ s ∣ α + f 2 ( x ) ∣ t ∣ β + g ( x )

for all ( x , s , t ) ∈ V × R × R ;

( F 3 ) ′ there are positive constants γ 1 , γ 2 , δ 1 , and δ 2 with δ i > γ i κ i , i = 1 , 2 , such that

Θ 1 ≔ 1 γ 1 p + γ 2 q max ∣ ( s , t ) ∣ ≤ 1 h 0 1 ⁄ p μ 0 1 ⁄ p ( p γ 1 p + p γ 2 q ) 1 p + 1 h 0 1 ⁄ q μ 0 1 ⁄ q ( q γ 1 p + q γ 2 q ) 1 q a ( ∣ ( s , t ) ∣ ) ∫ V b ( x ) d μ < F ( x 0 , δ 1 , δ 2 ) δ 1 p M 1 p + δ 2 q M 2 q ≔ Θ 2 ,

where κ 1 = M 1 p − 1 p , κ 2 = M 2 q − 1 q , and

M 1 = deg ( x 0 ) 2 μ ( x 0 ) p 2 μ ( x 0 ) + h 1 ( x 0 ) μ ( x 0 ) + ∑ y ∼ x 0 w x 0 y 2 μ ( y ) p 2 μ ( y ) , M 2 = deg ( x 0 ) 2 μ ( x 0 ) q 2 μ ( x 0 ) + h 2 ( x 0 ) μ ( x 0 ) + ∑ y ∼ x 0 w x 0 y 2 μ ( y ) q 2 μ ( y ) .

Then, for each parameter λ belonging to ( Θ 2 − 1 , Θ 1 − 1 ) , system (1.15) has at least three distinct solutions.

Remark 1.3

In Theorems 1.1 and 1.2, all three solutions are nontrivial solutions if we furthermore assume that F s ( x , 0 , 0 ) ≠ 0 or F t ( x , 0 , 0 ) ≠ 0 for some x ∈ V .

2 Preliminaries

In this section, we recall the Sobolev space on graph and some embedding relationships [2,7,10]. We also recall an abstract critical point theorem in [11], which is the main tool to prove our main results.

Let G = ( V , E ) be a graph. For any given integer m ≥ 1 and any given real number l > 1 , we define

W m , l ( V ) = u : V → R ∫ V ( ∣ ∇ m u ( x ) ∣ l + h ( x ) ∣ u ( x ) ∣ l ) d μ < ∞

endowed with the norm

‖ u ‖ W m , l ( V ) = ∫ V ( ∣ ∇ m u ( x ) ∣ l + h ( x ) ∣ u ( x ) ∣ l ) d μ 1 l ,

where h ( x ) > 0 for all x ∈ V . If V is a finite graph, then W m , l ( V ) is of finite dimension.

Lemma 2.1

[2,10] Let G = ( V , E ) be a finite graph. For all ψ ∈ W m , l ( V ) , there is

‖ ψ ‖ ∞ ≤ d l ‖ ψ ‖ W m , l ( V ) ,

where ‖ ψ ‖ ∞ = max x ∈ V ∣ ψ ( x ) ∣ and d l = 1 μ min h min 1 l .

Lemma 2.2

[2,10] Let G = ( V , E ) be a finite graph. Then, W m , l ( V ) ↪ L r ( V ) for all 1 ≤ r ≤ + ∞ . Particularly, if 1 < r < + ∞ , then for all ψ ∈ W m , l ( V ) ,

‖ ψ ‖ L r ( V ) ≤ K l , r ‖ ψ ‖ W m , l ( V ) ,

where

K l , r = ∑ x ∈ V μ ( x ) 1 r μ min 1 l h min 1 l .

Lemma 2.3

[7] Let G = ( V , E ) be a locally finite graph. If μ ( x ) ≥ μ 0 and ( H 1 ) holds, then W h 1 , l ( V ) is continuously embedded into L r ( V ) for all 1 < l ≤ r ≤ ∞ , and the following inequalities hold:

‖ u ‖ ∞ ≤ 1 h 0 1 ⁄ l μ 0 1 ⁄ l ‖ u ‖ W 1 , l ( V )

and

‖ u ‖ L r ( V ) ≤ μ 0 l − r l r h 0 − 1 l ‖ u ‖ W 1 , l ( V ) for all l ≤ r < ∞ .

Lemma 2.4

[11] Let W be a real reflexive Banach space, Φ : W → R be a coercive, continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on W * , Ψ : W → R be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that

Φ ( 0 ) = Ψ ( 0 ) = 0 .

Assume that there exist r > 0 and x ¯ ∈ X , with r < Φ ( x ¯ ) , such that:

  1. sup Φ ( x ) ≤ r Ψ ( x ) r < Ψ ( x ¯ ) Φ ( x ¯ ) ;

  2. for each λ ∈ Λ r ≔ Φ ( x ¯ ) Ψ ( x ¯ ) , r sup Φ ( x ) ≤ r Ψ ( x ) , the functional Φ − λ Ψ is coercive.

Then, for each λ ∈ Λ r , the functional φ λ ≔ Φ − λ Ψ has at least three distinct critical points in W.

3 Proofs for the poly-Laplacian system (1.14)

Let G = ( V , E ) be a finite graph. In order to investigate the poly-Laplacian system (1.14), we work in the space W ≔ W m 1 , p ( V ) × W m 2 , q ( V ) with the norm endowed with ‖ ( u , v ) ‖ = ‖ u ‖ W m 1 , p ( V ) + ‖ v ‖ W m 2 , q ( V ) . Then, ( W , ‖ ⋅ ‖ ) is a Banach space of finite dimension. Consider the functional φ : W → R as

(3.1) φ ( u , v ) = 1 p ∫ V ( ∣ ∇ m 1 u ∣ p + h 1 ( x ) ∣ u ∣ p ) d μ + 1 q ∫ V ( ∣ ∇ m 2 v ∣ q + h 2 ( x ) ∣ v ∣ q ) d μ − λ ∫ V F ( x , u , v ) d μ .

Then, under the assumptions of Theorem 1.1, φ ∈ C 1 ( W , R ) and

(3.2) ⟨ φ ′ ( u , v ) , ( ϕ 1 , ϕ 2 ) ⟩ = ∫ V [ £ m 1 , p u ϕ 1 + h 1 ( x ) ∣ u ∣ p − 2 u ϕ 1 − λ F u ( x , u , v ) ϕ 1 ] d μ + ∫ V [ £ m 2 , q v ϕ 2 + h 2 ( x ) ∣ v ∣ q − 2 v ϕ 2 − λ F v ( x , u , v ) ϕ 2 ] d μ

for any ( u , v ) , ( ϕ 1 , ϕ 2 ) ∈ W . In order to apply Lemma 2.4, we will use the functionals Φ : W → R and Ψ : W → R defined by setting

Φ ( u , v ) = 1 p ∫ V ( ∣ ∇ m 1 u ∣ p + h 1 ( x ) ∣ u ∣ p ) d μ + 1 q ∫ V ( ∣ ∇ m 2 v ∣ q + h 2 ( x ) ∣ v ∣ q ) d μ = 1 p ‖ u ‖ W m 1 , p ( V ) p + 1 q ‖ v ‖ W m 2 , q ( V ) q

and

Ψ ( u , v ) = ∫ V F ( x , u , v ) d μ .

Then, φ ( u , v ) = Φ − λ Ψ . Moreover, it is easy to see that ( u , v ) ∈ W is a critical point of φ if and only if

∫ V ( £ m 1 , p u + h 1 ( x ) ∣ u ∣ p − 2 u − λ F u ( x , u , v ) ) ϕ 1 d μ = 0

and

∫ V ( £ m 2 , q v + h 2 ( x ) ∣ v ∣ q − 2 v − λ F v ( x , u , v ) ) ϕ 2 d μ = 0

for all ( ϕ 1 , ϕ 2 ) ∈ W . By the arbitrariness of ϕ 1 and ϕ 2 , we conclude that

£ m 1 , p u + h 1 ( x ) ∣ u ∣ p − 2 u = λ F u ( x , u , v ) , £ m 2 , q v + h 2 ( x ) ∣ v ∣ q − 2 v = λ F v ( x , u , v ) .

Thus, the problem of finding the solutions for system (1.14) is reduced to seek the critical points of functional φ on W .

Lemma 3.1

Assume that ( F 0 ) holds. Then, for any given r > 0 , the following inequality holds:

sup ( u , v ) ∈ Φ − 1 ( − ∞ , r ] Ψ ( u , v ) r ≤ 1 r max x ∈ V , ∣ s ∣ ≤ ( p r ) 1 p h min 1 ⁄ p μ min 1 ⁄ p , ∣ t ∣ ≤ ( q r ) 1 q h min 1 ⁄ q μ min 1 ⁄ q F ( x , u ( x ) , v ( x ) ) ∣ V ∣ .

Proof

By ( F 0 ) , we have

Ψ ( u , v ) = ∫ V F ( x , u ( x ) , v ( x ) ) d μ = ∑ V F ( x , u ( x ) , v ( x ) ) d μ ≤ max x ∈ V , ∣ s ∣ ≤ ‖ u ‖ ∞ , ∣ t ∣ ≤ ‖ v ‖ ∞ F ( x , s , t ) ∣ V ∣

for every ( u , v ) ∈ W . Furthermore, for all ( u , v ) ∈ W with Φ ( u , v ) ≤ r , by Lemma 2.1, we obtain

‖ u ‖ ∞ ≤ 1 h 1 , min 1 ⁄ p μ min 1 ⁄ p ‖ u ‖ W m 1 , p ( V ) ≤ 1 h 1 , min 1 ⁄ p μ min 1 ⁄ p ( p r ) 1 p , ‖ v ‖ ∞ ≤ 1 h 2 , min 1 ⁄ q μ min 1 ⁄ q ‖ v ‖ W m 2 , q ( V ) ≤ 1 h 2 , min 1 ⁄ q μ min 1 ⁄ q ( q r ) 1 q .

Hence, we obtain

(3.3) sup ( u , v ) ∈ Φ − 1 ( − ∞ , r ] Ψ ( u , v ) ≤ max x ∈ V , ∣ s ∣ ≤ ( p r ) 1 p h 1 , min 1 ⁄ p μ min 1 ⁄ p , ∣ t ∣ ≤ ( q r ) 1 q h 2 , min 1 ⁄ q μ min 1 ⁄ q F ( x , s , t ) ∣ V ∣ .

Then, the proof is completed by multiplying 1 r on both sides of (3.3).□

Lemma 3.2

Assume that ( F 0 ) and ( F 3 ) hold. Then, there exists ( u δ 1 , v δ 2 ) ∈ W such that

sup ( u , v ) ∈ Φ − 1 ( − ∞ , γ 1 p + γ 2 q ] Ψ ( u , v ) γ 1 p + γ 2 q < Ψ ( u δ 1 , v δ 2 ) Φ ( u δ 1 , v δ 2 ) .

Proof

Let

u δ 1 ( x ) = δ 1 , v δ 2 ( x ) = δ 2 , ∀ x ∈ V ,

where δ i , i = 1 , 2 are given in ( F 3 ) . It is easy to verify that ( u δ 1 , v δ 2 ) ∈ W , ∣ ∇ m 1 u δ 1 ∣ = 0 , and ∣ ∇ m 1 v δ 1 ∣ = 0 for all m i ≥ 1 , i = 1 , 2 . Then,

(3.4) Φ ( u δ 1 , v δ 2 ) = 1 p ∫ V ( ∣ ∇ m 1 u δ 1 ∣ p + h 1 ( x ) ∣ u δ 1 ∣ p ) d μ + 1 q ∫ V ( ∣ ∇ m 2 v δ 2 ∣ q + h 2 ( x ) ∣ v δ 2 ∣ q ) d μ = δ 1 p p ∫ V h 1 ( x ) d μ + δ 2 q q ∫ V h 2 ( x ) d μ .

Note that δ i > γ i κ i , i = 1 , 2 , where

κ 1 = 1 p ∫ V h 1 ( x ) d μ − 1 p , κ 2 = 1 q ∫ V h 2 ( x ) d μ − 1 q .

Then, (3.4) implies that Φ ( u δ 1 , v δ 2 ) ≥ γ 1 p + γ 2 q . Moreover,

(3.5) Ψ ( u δ 1 , v δ 2 ) = ∫ V F ( x , u δ 1 , v δ 2 ) d μ = ∫ V F ( x , δ 1 , δ 2 ) d μ ≥ inf x ∈ V F ( x , δ 1 , δ 2 ) ∣ V ∣ .

Hence, by (3.4) and (3.5), we obtain

(3.6) Ψ ( u δ 1 , v δ 2 ) Φ ( u δ 1 , v δ 2 ) ≥ inf x ∈ V F ( x , δ 1 , δ 2 ) ∣ V ∣ δ 1 p p ∫ V h 1 ( x ) d μ + δ 2 q q ∫ V h 2 ( x ) d μ .

In view of Lemma 3.1, (3.6), and ( F 3 ), we obtain

sup ( u , v ) ∈ Φ − 1 ( − ∞ , γ 1 p + γ 2 q ] Ψ ( u , v ) γ 1 p + γ 2 q ≤ 1 γ 1 p + γ 2 q max x ∈ V , ∣ s ∣ ≤ 1 h min 1 ⁄ p μ min 1 ⁄ p ( p ( γ 1 p + γ 2 q ) ) 1 p , ∣ t ∣ ≤ 1 h min 1 ⁄ q μ min 1 ⁄ q ( q ( γ 1 p + γ 2 q ) ) 1 q F ( x , s , t ) ∣ V ∣ < inf x ∈ V F ( x , δ 1 , δ 2 ) ∣ V ∣ δ 1 p p ∫ V h 1 ( x ) d μ + δ 2 q q ∫ V h 2 ( x ) d μ ≤ Ψ ( u δ 1 , v δ 2 ) Φ ( u δ 1 , v δ 2 ) .

The proof is complete.□

Lemma 3.3

Assume that ( F 2 ) holds. Then, for each λ ∈ ( 0 , + ∞ ) , the functional Φ − λ Ψ is coercive.

Proof

By ( F 2 ) , we have

φ ( u , v ) = 1 p ‖ u ‖ W m 1 , p ( V ) p + 1 q ‖ v ‖ W m 2 , q ( V ) q − λ ∫ V F ( x , u , v ) d μ ≥ 1 p ‖ u ‖ W m 1 , p ( V ) p + 1 q ‖ v ‖ W m 2 , q ( V ) q − λ ‖ f 1 ‖ ∞ ‖ u ‖ ∞ α − λ ‖ f 2 ‖ ∞ ‖ v ‖ ∞ β − λ ∫ V g ( x ) d μ ≥ 1 p ‖ u ‖ W m 1 , p ( V ) p + 1 q ‖ v ‖ W m 2 , q ( V ) q − λ ‖ f 1 ‖ ∞ d p α ‖ u ‖ W m 1 , p ( V ) α − λ ‖ f 2 ‖ ∞ d q β ‖ v ‖ W m 2 , q ( V ) β − λ ∫ V g ( x ) d μ .

Note that α ∈ [ 0 , p ) and β ∈ [ 0 , q ) . Therefore, φ ( u , v ) is a coercive functional for every λ ∈ ( 0 , + ∞ ) .□

Lemma 3.4

The Gâteaux derivative of Φ admits a continuous inverse on W * , where W * is the dual space of W.

Proof

First, we prove that Φ ′ is uniformly monotone. Via ( 2.2 ) of [8], there exists a positive constant c p such that

(3.7) ( ∣ x ∣ p − 2 x − ∣ y ∣ p − 2 y , x − y ) ≥ c p ∣ x − y ∣ p , for all x , y ∈ R N ,

where ( ⋅ , ⋅ ) denotes the inner product in R N . Note that

⟨ Φ ′ ( u 1 , v 1 ) − Φ ′ ( u 2 , v 2 ) , ( u 1 − u 2 , v 1 − v 2 ) ⟩ = ∫ V [ £ m 1 , p u 1 ( u 1 − u 2 ) + h 1 ( x ) ∣ u 1 ∣ p − 2 u 1 ( u 1 − u 2 ) ] d μ + ∫ V [ £ m 2 , q v 1 ( v 1 − v 2 ) + h 2 ( x ) ∣ v 1 ∣ q − 2 v 1 ( v 1 − v 2 ) ] d μ − ∫ V [ £ m 1 , p u 2 ( u 1 − u 2 ) + h 1 ( x ) ∣ u 2 ∣ p − 2 u 2 ( u 1 − u 2 ) ] d μ − ∫ V [ £ m 2 , q v 2 ( v 1 − v 2 ) + h 2 ( x ) ∣ v 2 ∣ q − 2 v 2 ( v 1 − v 2 ) ] d μ .

First, we prove that

I ≔ ∫ V [ £ m 1 , p u 1 ( u 1 − u 2 ) + h 1 ( x ) ∣ u 1 ∣ p − 2 u 1 ( u 1 − u 2 ) − £ m 1 , p u 2 ( u 1 − u 2 ) + h 1 ( x ) ∣ u 2 ∣ p − 2 u 2 ( u 1 − u 2 ) ] d μ ≥ c p ‖ u 1 − u 2 ‖ W m 1 . p ( V ) p .

When m 1 is odd, by (1.13), (1.4), (1.7), (1.8), and (3.7), we have

∫ V [ £ m 1 , p u 1 ( u 1 − u 2 ) − £ m 1 , p u 2 ( u 1 − u 2 ) ] d μ = ∫ V ∣ ∇ m 1 u 1 ∣ p − 2 Γ Δ m 1 − 1 2 u 1 , Δ m 1 − 1 2 ( u 1 − u 2 ) − ∣ ∇ m 1 u 2 ∣ p − 2 Γ Δ m 1 − 1 2 u 2 , Δ m 1 − 1 2 ( u 1 − u 2 ) d μ = ∫ V ∣ ∇ m 1 u 1 ∣ p − 2 ∇ Δ m 1 − 1 2 u 1 ⋅ ∇ Δ m 1 − 1 2 ( u 1 − u 2 ) − ∣ ∇ m 1 u 2 ∣ p − 2 ∇ Δ m 1 − 1 2 u 2 ⋅ ∇ Δ m 1 − 1 2 ( u 1 − u 2 ) d μ = ∫ V ∣ ∇ Δ m 1 − 1 2 u 1 ∣ p − 2 ∇ Δ m 1 − 1 2 u 1 ⋅ ∇ Δ m 1 − 1 2 ( u 1 − u 2 ) − ∣ ∇ Δ m 1 − 1 2 u 2 ∣ p − 2 ∇ Δ m 1 − 1 2 u 2 ⋅ ∇ Δ m 1 − 1 2 ( u 1 − u 2 ) d μ = ∫ V ∇ Δ m 1 − 1 2 ( u 1 − u 2 ) ⋅ ( ∣ ∇ Δ m 1 − 1 2 u 1 ∣ p − 2 ∇ Δ m 1 − 1 2 u 1 − ∣ ∇ Δ m 1 − 1 2 u 2 ∣ p − 2 ∇ Δ m 1 − 1 2 u 2 ) d μ ≥ ∫ V c p ∇ Δ m 1 − 1 2 ( u 1 − u 2 ) p d μ = ∫ V c p ∣ ∇ m 1 ( u 1 − u 2 ) ∣ p d μ .

When m 1 is even, by (1.13), (1.7), (1.8), and (3.7), we also have

∫ V [ £ m 1 , p u 1 ( u 1 − u 2 ) − £ m 1 , p u 2 ( u 1 − u 2 ) ] d μ = ∫ V ∣ ∇ m 1 u 1 ∣ p − 2 Δ m 1 2 u 1 Δ m 1 2 ( u 1 − u 2 ) − ∣ ∇ m 1 u 2 ∣ p − 2 Δ m 1 2 u 2 Δ m 1 2 ( u 1 − u 2 ) d μ = ∫ V Δ m 1 2 u 1 p − 2 Δ m 1 2 u 1 Δ m 1 2 ( u 1 − u 2 ) − Δ m 1 2 u 2 p − 2 Δ m 1 2 u 2 Δ m 1 2 ( u 1 − u 2 ) d μ = ∫ V Δ m 1 2 ( u 1 − u 2 ) ( Δ m 1 2 u 1 p − 2 Δ m 1 2 u 1 − Δ m 1 2 u 2 p − 2 Δ m 1 2 u 2 ) d μ ≥ ∫ V c p Δ m 1 2 ( u 1 − u 2 ) p d μ = ∫ V c p ∣ ∇ m 1 ( u 1 − u 2 ) ∣ p d μ .

Thus, for all positive integers m , we obtain

(3.8) ∫ V [ £ m 1 , p u 1 ( u 1 − u 2 ) − £ m 1 , p u 2 ( u 1 − u 2 ) ] d μ ≥ ∫ V c p ∣ ∇ m 1 ( u 1 − u 2 ) ∣ p d μ .

By (3.7), we have

(3.9) ∫ V [ h 1 ( x ) ∣ u 1 ∣ p − 2 u 1 ( u 1 − u 2 ) − h 1 ( x ) ∣ u 2 ∣ p − 2 u 2 ( u 1 − u 2 ) ] d μ ≥ ∫ V h 1 ( x ) c p ∣ u 1 − u 2 ∣ p d μ .

So, by (3.8) and (3.9), we obtain that

I ≥ c p ‖ u 1 − u 2 ‖ W m 1 . p ( V ) p .

Similarly, we can prove that there exists a positive constant c q such that

I I ≔ ∫ V [ £ m 2 , q v 1 ( v 1 − v 2 ) + h 2 ( x ) ∣ v 1 ∣ q − 2 v 1 ( v 1 − v 2 ) − £ m 2 , q v 2 ( v 1 − v 2 ) + h 2 ( x ) ∣ v 2 ∣ q − 2 v 2 ( v 1 − v 2 ) ] d μ ≥ c q ‖ v 1 − v 2 ‖ W m 2 . q ( V ) q .

Hence,

(3.10) ⟨ Φ ′ ( u 1 , v 1 ) − Φ ′ ( u 2 , v 2 ) , ( u 1 − u 2 , v 1 − v 2 ) ⟩ ≥ c p ‖ u 1 − u 2 ‖ W m 1 . p ( V ) p + c q ‖ v 1 − v 2 ‖ W m 2 . q ( V ) q .

Next we consider the following four cases if we let max { p , q } = p .

(1) Assume that ‖ u 1 − u 2 ‖ W m 1 . p ( V ) > 1 and ‖ v 1 − v 2 ‖ W m 2 . q ( V ) > 1 . Then, ‖ ( u 1 − u 2 , v 1 − v 2 ) ‖ > 2 and

(3.11) c p ‖ u 1 − u 2 ‖ W m 1 . p ( V ) p + c q ‖ v 1 − v 2 ‖ W m 2 . q ( V ) q ≥ min { c p , c q } ( ‖ u 1 − u 2 ‖ W m 1 . p ( V ) q + ‖ v 1 − v 2 ‖ W m 2 . q ( V ) q ) ≥ min { c p , c q } 2 q − 1 ‖ ( u 1 − u 2 , v 1 − v 2 ) ‖ q ≥ min { c p , c q } 2 p − 1 ‖ ( u 1 − u 2 , v 1 − v 2 ) ‖ q .

Let

(3.12) a 1 ( t ) = min { c p , c q } 2 p − 1 t q − 1 , t > 2 .

(2) Assume that ‖ u 1 − u 2 ‖ W m 1 . p ( V ) ≤ 1 and ‖ v 1 − v 2 ‖ W m 2 . q ( V ) ≤ 1 . Then, ‖ ( u 1 − u 2 , v 1 − v 2 ) ‖ ≤ 2 and

(3.13) c p ‖ u 1 − u 2 ‖ W m 1 . p ( V ) p + c q ‖ v 1 − v 2 ‖ W m 2 . q ( V ) q ≥ min { c p , c q } ( ‖ u 1 − u 2 ‖ W m 1 . p ( V ) p + ‖ v 1 − v 2 ‖ W m 2 . q ( V ) p ) ≥ min { c p , c q } 2 p − 1 ‖ ( u 1 − u 2 , v 1 − v 2 ) ‖ p ≥ min { c p , c q } 2 p − 1 ‖ ( u 1 − u 2 , v 1 − v 2 ) ‖ p , ‖ ( u 1 − u 2 , v 1 − v 2 ) ‖ ≤ 1 , min { c p , c q } 2 p − 1 ‖ ( u 1 − u 2 , v 1 − v 2 ) ‖ q , 1 < ‖ ( u 1 − u 2 , v 1 − v 2 ) ‖ ≤ 2 .

Let

(3.14) a 2 ( t ) = min { c p , c q } 2 p − 1 t p − 1 , 0 ≤ t ≤ 1 , min { c p , c q } 2 p − 1 t q − 1 , 1 < t ≤ 2 .

(3) Assume that ‖ u 1 − u 2 ‖ W m 1 . p ( V ) > 1 and ‖ v 1 − v 2 ‖ W m 2 . q ( V ) ≤ 1 . Then, ‖ ( u 1 − u 2 , v 1 − v 2 ) ‖ q > 1 and

(3.15) c p ‖ u 1 − u 2 ‖ W m 1 . p ( V ) p + c q ‖ v 1 − v 2 ‖ W m 2 . q ( V ) q ≥ min { c p , c q } ( ‖ u 1 − u 2 ‖ W m 1 . p ( V ) q + ‖ v 1 − v 2 ‖ W m 2 . q ( V ) q ) ≥ min { c p , c q } 2 q − 1 ‖ ( u 1 − u 2 , v 1 − v 2 ) ‖ q ≥ min { c p , c q } 2 p − 1 ‖ ( u 1 − u 2 , v 1 − v 2 ) ‖ q .

Let

(3.16) a 3 ( t ) = min { c p , c q } 2 p − 1 t q − 1 , t > 1 .

(4) Assume that ‖ u 1 − u 2 ‖ W m 1 . p ( V ) ≤ 1 and ‖ v 1 − v 2 ‖ W m 2 . q ( V ) > 1 . Then, ‖ ( u 1 − u 2 , v 1 − v 2 ) ‖ q > 1 . Note that q − p ≤ 0 . Thus, we have

(3.17) c p ‖ u 1 − u 2 ‖ W m 1 . p ( V ) p + c q ‖ v 1 − v 2 ‖ W m 2 . q ( V ) q ≥ min { c p , c q ‖ v 1 − v 2 ‖ W m 2 . q ( V ) q − p } ( ‖ u 1 − u 2 ‖ W m 1 . p ( V ) p + ‖ v 1 − v 2 ‖ W m 2 . q ( V ) p ) ≥ min { c p , c q ( ‖ u 1 − u 2 ‖ W m 1 . p ( V ) + ‖ v 1 − v 2 ‖ W m 2 . q ( V ) ) q − p } ( ‖ u 1 − u 2 ‖ W m 1 . p ( V ) p + ‖ v 1 − v 2 ‖ W m 2 . q ( V ) p ) ≥ min { c p , c q ‖ ( u 1 − u 2 , v 1 − v 2 ) ‖ q − p } 1 2 p − 1 ‖ ( u 1 − u 2 , v 1 − v 2 ) ‖ p = min c p 2 p − 1 ‖ ( u 1 − u 2 , v 1 − v 2 ) ‖ p , c q 2 p − 1 ‖ ( u 1 − u 2 , v 1 − v 2 ) ‖ q ≥ min { c p , c q } 2 p − 1 ‖ ( u 1 − u 2 , v 1 − v 2 ) ‖ q .

Let

(3.18) a 4 ( t ) = min { c p , c q } 2 p − 1 t q − 1 , t > 1 .

Combining (3.12), (3.14), (3.16), and (3.18), we define a : R + ∪ { 0 } → R + ∪ { 0 } by

(3.19) a ( t ) = min { c p , c q } 2 p − 1 t p − 1 , 0 ≤ t ≤ 1 , min { c p , c q } 2 p − 1 t q − 1 , t > 1 .

Then, a is continuous and strictly monotone increasing with a ( 0 ) = 0 and a ( t ) → + ∞ as t → + ∞ . Thus, by (3.11), (3.13), (3.15), and (3.17), (3.10) can be written as

⟨ Φ ′ ( u 1 , v 1 ) − Φ ′ ( u 2 , v 2 ) , ( u 1 − u 2 , v 1 − v 2 ) ⟩ ≥ a ( ‖ ( u 1 − u 2 , v 1 − v 2 ) ‖ W ) ‖ ( u 1 − u 2 , v 1 − v 2 ) ‖ W .

So Φ ′ is uniformly monotone in W if max { p , q } = p . Similarly, if we let max { p , q } = q , we can also obtain the same conclusion.

Next we show that Φ ′ is also hemicontinuous in W . Assume that s → s * and s , s * ∈ [ 0 , 1 ] . Note that

∣ ⟨ Φ ′ ( ( u 1 , u 2 ) + s ( v 1 , v 2 ) ) , ( w 1 , w 2 ) ⟩ − ⟨ Φ ′ ( ( u 1 , u 2 ) + s * ( v 1 , v 2 ) ) , ( w 1 , w 2 ) ⟩ ∣ ≤ ‖ Φ ′ ( ( u 1 , u 2 ) + s ( v 1 , v 2 ) ) − Φ ′ ( ( u 1 , u 2 ) + s * ( v 1 , v 2 ) ) ‖ * ⋅ ‖ ( w 1 , w 2 ) ‖

for all ( u 1 , u 2 ) , ( v 1 , v 2 ) , ( w 1 , w 2 ) ∈ W , where ‖ ⋅ ‖ * denotes the norm of the dual space W * . Then, the continuity of Φ ′ implies that

⟨ Φ ′ ( ( u 1 , u 2 ) + s ( v 1 , v 2 ) ) , ( w 1 , w 2 ) ⟩ → ⟨ Φ ′ ( ( u 1 , u 2 ) + s * ( v 1 , v 2 ) ) , ( w 1 , w 2 ) ⟩ , as s → s *

for all ( u 1 , u 2 ) , ( v 1 , v 2 ) , ( w 1 , w 2 ) ∈ W . Hence Φ ′ is hemicontinuous in W .

Moreover, for all ( u , v ) ∈ W , we have

⟨ Φ ′ ( u , v ) , ( u , v ) ⟩ = ∫ V [ ∣ ∇ m 1 u ∣ p + h 1 ( x ) ∣ u ∣ p ] d μ + ∫ V [ ∣ ∇ m 2 v ∣ q + h 2 ( x ) ∣ v ∣ q ] d μ = ‖ u ‖ W m 1 , p ( V ) p + ‖ v ‖ W m 2 , q ( V ) q .

So, Φ ′ is coercive in W . Thus, by Theorem 26.A in [13], we can obtain that Φ ′ admits a continuous inverse in W .□

Lemma 3.5

Φ : W → R is sequentially weakly lower semi-continuous.

Proof

Since Φ is continuously differentiable and Φ ′ is uniformly monotone, which implies that Φ ′ is monotone. It follows from Proposition 25.20 in [13] that Φ is sequentially weakly lower semi-continuous.□

Lemma 3.6

Ψ has compact derivative.

Proof

Obviously, Ψ is a C 1 functional on W . Assume that { ( u n , v n ) } ⊂ W is bounded. Note that W is of finite dimension. Then, there exists a subsequence { ( u k , v k ) } such that ( u k , v k ) → ( u 0 , v 0 ) for some ( u 0 , v 0 ) ∈ W . By the continuity of Ψ ′ , it is easy to obtain that

‖ Ψ ′ ( u k , v k ) − Ψ ′ ( u 0 , v 0 ) ‖ * → 0 , as n → ∞ .

Hence, Ψ ′ is compact in W .□

Proof of Theorem 1.1

Obviously, by ( F 0 ) and ( F 1 ) , Φ ( 0 ) = Ψ ( 0 ) = 0 and both Φ and Ψ are continuously differentiable. Moreover, it is easy to see that Φ : W → R is coercive. Lemmas 3.2–3.6 imply that all other conditions in Lemma 2.4 are satisfied. Hence, Lemma 2.4 implies that for each λ ∈ ( Λ 2 − 1 , Λ 1 − 1 ) , the functional φ has at least three distinct critical points that are solutions of system (1.14).□

4 Proofs for the ( p , q ) -Laplacian system (1.15)

Let G = ( V , E ) be a locally finite graph. In order to investigate the ( p , q ) -Laplacian system (1.15), we work in the space W 1 ≔ W 1 , p ( V ) × W 1 , q ( V ) with the norm endowed with ‖ ( u , v ) ‖ 1 = ‖ u ‖ W 1 , p ( V ) + ‖ v ‖ W 1 , q ( V ) and then, ( W 1 , ‖ ⋅ ‖ 1 ) is a Banach space which is of infinite dimension. Different from the case of finite graph in Section 2, the continuous differentiability of variational functional for (1.15) cannot be obtained just by ( F 0 ) . However, by using the condition ( F 0 ) ′ , the difficulty has been overcome in [7] so that we can apply Lemma 2.4 to system (1.15).

We consider the functional φ ¯ : W 1 → R as

(4.1) φ ¯ ( u , v ) = 1 p ∫ V ( ∣ ∇ u ∣ p + h 1 ( x ) ∣ u ∣ p ) d μ + 1 q ∫ V ( ∣ ∇ v ∣ q + h 2 ( x ) ∣ v ∣ q ) d μ − λ ∫ V F ( x , u , v ) d μ .

Then, by Appendix A.2 in [7], under the assumptions of Theorem 1.2, we have φ ¯ ∈ C 1 ( W 1 , R ) , and

(4.2) ⟨ φ ¯ ′ ( u , v ) , ( ϕ 1 , ϕ 2 ) ⟩ = ∫ V [ ∣ ∇ u ∣ p − 2 Γ ( u , ϕ 1 ) + h 1 ( x ) ∣ u ∣ p − 2 u ϕ 1 − λ F u ( x , u , v ) ϕ 1 ] d μ + ∫ V [ ∣ ∇ v ∣ q − 2 Γ ( v , ϕ 2 ) + h 2 ( x ) ∣ v ∣ q − 2 v ϕ 2 − λ F v ( x , u , v ) ϕ 2 ] d μ

for any ( u , v ) , ( ϕ 1 , ϕ 2 ) ∈ W 1 . Define Φ ¯ : W 1 → R and Ψ ¯ : W 1 → R by

Φ ¯ ( u , v ) = 1 p ∫ V ( ∣ ∇ u ∣ p + h 1 ( x ) ∣ u ∣ p ) d μ + 1 q ∫ V ( ∣ ∇ v ∣ q + h 2 ( x ) ∣ v ∣ q ) d μ = 1 p ‖ u ‖ W 1 , p ( V ) p + 1 q ‖ v ‖ W 1 , q ( V ) q

and

Ψ ¯ ( u , v ) = ∫ V F ( x , u , v ) d μ .

Then, φ ¯ ( u , v ) = Φ ¯ − λ Ψ ¯ . Moreover, it is easy to see that ( u , v ) ∈ W 1 is a critical point of φ ¯ if and only if

∫ V [ ∣ ∇ u ∣ p − 2 Γ ( u , ϕ 1 ) + h 1 ( x ) ∣ u ∣ p − 2 u ϕ 1 − λ F u ( x , u , v ) ϕ 1 ] d μ = 0

and

∫ V [ ∣ ∇ v ∣ q − 2 Γ ( v , ϕ 2 ) + h 2 ( x ) ∣ v ∣ q − 2 v ϕ 2 − λ F v ( x , u , v ) ϕ 2 ] d μ = 0

for all ( ϕ 1 , ϕ 2 ) ∈ W 1 .

Lemma 4.1

Assume that ( M ) , ( H 1 ) , and ( F 0 ) ′ hold. Then, for any given r > 0 , the following inequality holds:

sup ( u , v ) ∈ Φ ¯ − 1 ( − ∞ , r ] Ψ ¯ ( u , v ) r ≤ 1 r max ∣ ( s , t ) ∣ ≤ 1 h 0 1 ⁄ p μ 0 1 ⁄ p ( p r ) 1 p + 1 h 0 1 ⁄ q μ 0 1 ⁄ q ( q r ) 1 q a ( ∣ ( s , t ) ∣ ) ∫ V b ( x ) d μ .

Proof

By ( F 0 ) ′ , we have

Ψ ¯ ( u , v ) = ∫ V F ( x , u ( x ) , v ( x ) ) d μ ≤ ∫ V a ( ∣ ( u ( x ) , v ( x ) ) ∣ ) b ( x ) d μ ≤ max ∣ ( s , t ) ∣ ≤ ‖ u ‖ ∞ + ‖ v ‖ ∞ a ( ∣ ( s , t ) ∣ ) ∫ V b ( x ) d μ

for every ( u , v ) ∈ W 1 . Furthermore, for all ( u , v ) ∈ W 1 with Φ ¯ ( u , v ) ≤ r , by Lemma 2.3, we obtain

‖ u ‖ ∞ ≤ 1 h 0 1 ⁄ p μ 0 1 ⁄ p ‖ u ‖ W 1 , p ( V ) ≤ 1 h 0 1 ⁄ p μ 0 1 ⁄ p ( p r ) 1 p , ‖ v ‖ ∞ ≤ 1 h 0 1 ⁄ q μ 0 1 ⁄ q ‖ v ‖ W 1 , q ( V ) ≤ 1 h 0 1 ⁄ q μ 0 1 ⁄ q ( q r ) 1 q .

Then,

(4.3) sup ( u , v ) ∈ Φ ¯ − 1 ( − ∞ , r ] Ψ ¯ ( u , v ) ≤ max ∣ ( s , t ) ∣ ≤ 1 h 0 1 ⁄ p μ 0 1 ⁄ p ( p r ) 1 p + 1 h 0 1 ⁄ q μ 0 1 ⁄ q ( q r ) 1 q a ( ∣ ( s , t ) ∣ ) ∫ V b ( x ) d μ .

Furthermore, the proof is completed by multiplying 1 r on both sides of (4.3).□

Lemma 4.2

Assume that ( M ) , ( H 1 ) , ( F 0 ) ′ , and ( F 3 ) ′ hold. Then, there exists ( u δ 1 , v δ 2 ) ∈ W 1 such that

sup ( u , v ) ∈ Φ ¯ − 1 ( − ∞ , γ 1 p + γ 2 q ] Ψ ¯ ( u , v ) γ 1 p + γ 2 q < Ψ ¯ ( u δ 1 , v δ 2 ) Φ ¯ ( u δ 1 , v δ 2 ) .

Proof

Let

u δ 1 ( x ) = δ 1 , x = x 0 0 , x ≠ x 0 , v δ 2 ( x ) = δ 2 , x = x 0 0 , x ≠ x 0 ,

where δ i , i = 1 , 2 are defined in ( F 3 ) ′ . Then, a simple calculation implies that

∣ ∇ u δ 1 ∣ ( x ) = deg ( x 0 ) 2 μ ( x 0 ) δ 1 , x = x 0 , w x 0 y 2 μ ( y ) δ 1 , x = y with y ∼ x 0 , 0 , otherwise ,

and

∣ ∇ v δ 2 ∣ ( x ) = deg ( x 0 ) 2 μ ( x 0 ) δ 2 , x = x 0 , w x 0 y 2 μ ( y ) δ 2 , x = y with y ∼ x 0 , 0 , otherwise .

Then,

(4.4) ∫ V ( ∣ ∇ u δ 1 ∣ p + h 1 ( x ) ∣ u δ 1 ∣ p ) d μ = ∑ x ∈ V ( ∣ ∇ u δ 1 ∣ p + h 1 ( x ) ∣ u δ 1 ∣ p ) μ ( x ) = ( ∣ ∇ u δ 1 ∣ p ( x 0 ) + h 1 ( x 0 ) ∣ u δ 1 ∣ p ( x 0 ) ) μ ( x 0 ) + ∑ y ∼ x 0 ( ∣ ∇ u δ 1 ∣ p ( y ) + h 1 ( y ) ∣ u δ 1 ∣ p ( y ) ) μ ( y ) = deg ( x 0 ) 2 μ ( x 0 ) p 2 δ 1 p μ ( x 0 ) + h 1 ( x 0 ) δ 1 p μ ( x 0 ) + δ 1 p ∑ y ∼ x 0 w x 0 y 2 μ ( y ) p 2 μ ( y ) = δ 1 p M 1 ( defined in ( F 3 ) ′ ) ,

and similarly,

(4.5) ∫ V ( ∣ ∇ v δ 2 ∣ q + h 2 ( x ) ∣ v δ 2 ∣ q ) d μ = deg ( x 0 ) 2 μ ( x 0 ) q 2 δ 2 q μ ( x 0 ) + h 2 ( x 0 ) δ 2 q μ ( x 0 ) + δ 2 q ∑ y ∼ x 0 w x 0 y 2 μ ( y ) q 2 μ ( y ) = δ 2 q M 2 ( defined in ( F 3 ) ′ ) .

Note that { y ∣ y ∼ x 0 } is a finite set. Then, (4.4) and (4.5) imply that ( u δ 1 , v δ 2 ) ∈ W 1 . Moreover,

(4.6) Φ ¯ ( u δ 1 , v δ 2 ) = 1 p ∫ V ( ∣ ∇ u δ 1 ∣ p + h 1 ( x ) ∣ u δ 1 ∣ p ) d μ + 1 q ∫ V ( ∣ ∇ v δ 2 ∣ q + h 2 ( x ) ∣ v δ 2 ∣ q ) d μ = δ 1 p M 1 p + δ 2 q M 2 q .

Note that δ i > γ i κ i , i = 1 , 2 , where

κ 1 = M 1 p − 1 p , κ 2 = M 2 q − 1 q .

Then, (4.6) implies that Φ ¯ ( u δ 1 , v δ 2 ) ≥ γ 1 p + γ 2 q . Moreover, ( F 1 ) ′ implies that

(4.7) Ψ ¯ ( u δ 1 , v δ 2 ) = ∫ V F ( x , u δ 1 , v δ 2 ) d μ = F ( x 0 , δ 1 , δ 2 ) + ∫ V ⁄ { x 0 } F ( x , 0 , 0 ) d μ = F ( x 0 , δ 1 , δ 2 ) .

Hence, by (4.6) and (4.7), we obtain

Ψ ¯ ( u δ 1 , v δ 2 ) Φ ¯ ( u δ 1 , v δ 2 ) = F ( x 0 , δ 1 , δ 2 ) δ 1 p M 1 p + δ 2 q M 2 q .

In view of Lemma 4.1 and ( F 3 ) ′ , we obtain

sup ( u , v ) ∈ Φ ¯ − 1 ( − ∞ , γ 1 p + γ 2 q ] Ψ ¯ ( u , v ) γ 1 p + γ 2 q ≤ 1 γ 1 p + γ 2 q max ∣ ( s , t ) ∣ ≤ 1 h 0 1 ⁄ p μ 0 1 ⁄ p ( p γ 1 p + p γ 2 q ) 1 p + 1 h 0 1 ⁄ q μ 0 1 ⁄ q ( q γ 1 p + q γ 2 q ) 1 q a ( ∣ ( s , t ) ∣ ) ∫ V b ( x ) d μ < F ( x 0 , δ 1 , δ 2 ) δ 1 p M 1 p + δ 2 q M 2 q = Ψ ¯ ( u δ 1 , v δ 2 ) Φ ¯ ( u δ 1 , v δ 2 ) .

The proof is complete.□

Lemma 4.3

Assume that ( H 1 ) and ( F 2 ) ′ hold. Then, for each λ ∈ ( 0 , + ∞ ) , the functional Φ ¯ − λ Ψ ¯ is coercive.

Proof

By ( F ¯ 2 ) ′ , we have

φ ( u , v ) = 1 p ‖ u ‖ W 1 , p ( V ) p + 1 q ‖ v ‖ W 1 , q ( V ) q − λ ∫ V F ( x , u , v ) d μ ≥ 1 p ‖ u ‖ W 1 , p ( V ) p + 1 q ‖ v ‖ W 1 , q ( V ) q − λ ‖ f 1 ‖ ∞ ‖ u ‖ ∞ α − λ ‖ f 2 ‖ ∞ ‖ v ‖ ∞ β − λ ∫ V g ( x ) d μ ≥ 1 p ‖ u ‖ W 1 , p ( V ) p + 1 q ‖ v ‖ W 1 , q ( V ) q − λ ‖ f 1 ‖ ∞ h 0 − α p μ 0 − α p ‖ u ‖ W 1 , p ( V ) α − λ ‖ f 2 ‖ ∞ h 0 − β q μ 0 − β q ‖ v ‖ W 1 , q ( V ) β − λ ∫ V g ( x ) d μ .

Note that α ∈ [ 0 , p ) and β ∈ [ 0 , q ) . Therefore, φ ¯ ( u , v ) is a coercive functional for every λ ∈ ( 0 , + ∞ ) .□

Lemma 4.4

The Gâteaux derivative of Φ ¯ admits a continuous inverse on W 1 * , where W 1 * is the dual space of W 1 .

Proof

In the proof of Lemma 3.4, we only need to take m i = 1 , i = 1 , 2 and let G = ( V , E ) be a locally finite graph. Then, the proof is essentially the same as that in Lemma 3.4.□

Lemma 4.5

Φ ¯ : W 1 → R is sequentially weakly lower semi-continuous.

Proof

The proof is essentially the same as that in Lemma 3.5, taking m i = 1 , i = 1 , 2 and letting G = ( V , E ) be a locally finite graph.□

Lemma 4.6

Ψ ¯ has compact derivative.

Proof

Obviously, Ψ ¯ is a C 1 functional on W 1 . Assume that { ( u n , v n ) } ⊂ W 1 is bounded. Then, by Lemma 2.3, there exists a positive constant M > 0 such that ‖ u k ‖ ∞ ≤ M and ‖ v k ‖ ∞ ≤ M and there exists a subsequence { ( u k , v k ) } such that ( u k , v k ) ⇀ ( u 0 , v 0 ) for some ( u 0 , v 0 ) ∈ W 1 . In particular,

lim k → ∞ ∫ V u k φ d μ = ∫ V u 0 φ d μ , ∀ φ ∈ C c ( V ) ,

which implies that

(4.8) lim k → ∞ u k ( x ) = u 0 ( x ) for any fixed x ∈ V

by taking

φ ( y ) = 1 , y = x 0 , y ≠ x .

Similarly, we have

(4.9) lim k → ∞ v k ( x ) = v 0 ( x ) for any fixed x ∈ V .

Note that

‖ Ψ ¯ ′ ( u k , v k ) − Ψ ¯ ′ ( u 0 , v 0 ) ‖ * = sup ‖ ( ϕ 1 , ϕ 2 ) ‖ = 1 ∣ ⟨ Ψ ¯ ′ ( u k , v k ) − Ψ ¯ ′ ( u 0 , v 0 ) , ( ϕ 1 , ϕ 2 ) ⟩ ∣ = sup ‖ ( ϕ 1 , ϕ 2 ) ‖ = 1 ∫ V [ F u k ( x , u k , v k ) − F u 0 ( x , u 0 , v 0 ) ] ϕ 1 d μ + ∫ V [ F v k ( x , u k , v k ) − F v 0 ( x , u 0 , v 0 ) ] ϕ 2 d μ ≤ sup ‖ ( ϕ 1 , ϕ 2 ) ‖ = 1 ∫ V [ F u k ( x , u k , v k ) − F u 0 ( x , u 0 , v 0 ) ] ϕ 1 d μ + ∫ V [ F v k ( x , u k , v k ) − F v 0 ( x , u 0 , v 0 ) ] ϕ 2 d μ .

By ( F 0 ) ′ , we have

∣ F u k ( x , u k , v k ) − F u 0 ( x , u 0 , v 0 ) ∣ ≤ [ a ( ∣ ( u k , v k ) ∣ ) + a ( ∣ ( u 0 , v 0 ) ∣ ) ] b ( x ) ≤ [ max ∣ ( s , t ) ∣ ≤ ‖ u 0 ‖ ∞ + ‖ v 0 ‖ ∞ a ( ∣ ( s , t ) ∣ ) + max ∣ ( s , t ) ∣ ≤ 2 M a ( ∣ ( s , t ) ∣ ) ] b ( x ) ≔ l ( x ) .

Note that b ∈ L 1 ( V ) . Hence, l ( x ) ∈ L 1 ( V ) and so ∫ V ∣ F u k ( x , u k , v k ) − F u 0 ( x , u 0 , v 0 ) ∣ d μ is uniformly convergent. Thus, by (4.8), (4.9), and the continuity of F u , we have

∫ V [ F u k ( x , u k , v k ) − F u 0 ( x , u 0 , v 0 ) ] ϕ 1 d μ ≤ ∫ V ∣ F u k ( x , u k , v k ) − F u 0 ( x , u 0 , v 0 ) ∣ d μ ‖ ϕ 1 ‖ ∞ ≤ ∫ V ∣ F u k ( x , u k , v k ) − F u 0 ( x , u 0 , v 0 ) ∣ d μ h 0 − 1 p μ 0 − 1 p → 0 as k → ∞ .

Similarly, we also have

∫ V [ F v k ( x , u k , v k ) − F v 0 ( x , u 0 , v 0 ) ] ϕ 2 d μ → 0 as k → ∞ .

So,

‖ Ψ ¯ ′ ( u k , v k ) − Ψ ¯ ′ ( u 0 , v 0 ) ‖ * → 0 as k → ∞ .

Hence, Ψ ¯ ′ is compact in W 1 .□

Proof of Theorem 1.2

Obviously, by ( F 0 ) ′ and ( F 1 ) ′ , Φ ¯ ( 0 ) = Ψ ¯ ( 0 ) = 0 and both Φ ¯ and Ψ ¯ are continuously differentiable. Moreover, it is easy to see that Φ ¯ : W 1 → R is coercive. Lemmas 4.2–4.6 imply that all other conditions in Lemma 2.4 are satisfied. Hence, Lemma 2.4 implies that for each λ ∈ ( Θ 2 − 1 , Θ 1 − 1 ) , the functional φ ¯ has at least three distinct critical points that are solutions of system (1.15).□

Remark 4.7

On the locally finite graph, we do not consider the more general poly-Laplacian system. That is because it is difficult to obtain the continuous differentiability of the variational functional φ when m i > 1 , i = 1 , 2 , which is caused by the special definition of ℒ m , p .

5 Results of the scalar equation

By using the similar arguments of Theorem 1.1, we can also obtain a similar result for the following scalar equation on finite graph G = ( V , E ) :

(5.1) £ m , p u + h ( x ) ∣ u ∣ p − 2 u = λ f ( x , u ) , x ∈ V ,

where m ≥ 1 is an integer, h : V → R , p > 1 , λ > 0 , and f : V × R → R .

Theorem 5.1

Let G = ( V , E ) be a finite graph and F ( x , s ) = ∫ 0 s f ( x , τ ) d τ for all x ∈ V . Assume that the following conditions hold:

  1. h ( x ) > 0 for all x ∈ V ;

  2. F ( x , s ) is continuously differentiable in s ∈ R for all x ∈ V ;

  3. ∫ V F ( x , 0 ) d μ = 0 ;

  4. there exist a constant α ∈ [ 0 , p ) and functions g 1 , g 2 : V → R such that

    F ( x , s ) ≤ g 1 ( x ) ∣ s ∣ α + g 2 ( x )

    for all ( x , s ) ∈ V × R ;

  5. there are positive constants γ and δ with δ > γ κ , such that

    Λ 1 ′ ≔ 1 γ p max x ∈ V , ∣ s ∣ ≤ ( p γ p ) 1 p h min 1 ⁄ p μ min 1 ⁄ p F ( x , s ) ∣ V ∣ < inf x ∈ V F ( x , δ ) ∣ V ∣ δ p p ∫ V h ( x ) d μ ≔ Λ 2 ′ ,

    where ∣ V ∣ = ∑ x ∈ V μ ( x ) and κ = 1 p ∫ V h ( x ) d μ − 1 p .

Then, for each parameter λ belonging to ( Λ 2 ′ − 1 , Λ 1 ′ − 1 ) , equation (5.1) has at least three distinct solutions.

By using similar arguments of Theorem 1.2, we can also obtain a similar result for the following scalar equation on a locally finite graph G = ( V , E ) :

(5.2) − Δ p u + h ( x ) ∣ u ∣ p − 2 u = λ f ( x , u ) , x ∈ V ,

where p ≥ 2 , h : V → R , λ > 0 , and f : V × R → R .

Theorem 5.2

Let G = ( V , E ) be a locally finite graph and F ( x , s ) = ∫ 0 s f ( x , τ ) d τ for all x ∈ V . Assume that ( M ) and the following conditions hold:

( h ) ′ there exist a constant h 0 > 0 such that h ( x ) ≥ h 0 > 0 for all x ∈ V ;

( f 0 ) ′ F ( x , s ) is continuously differentiable in s ∈ R for all x ∈ V , and there exist a function a ∈ C ( R + , R + ) and a function b : V → R + with b ∈ L 1 ( V ) such that

∣ f ( x , s ) ∣ ≤ a ( ∣ s ∣ ) b ( x ) , ∣ F ( x , s ) ∣ ≤ a ( ∣ s ∣ ) b ( x ) ,

for all x ∈ V and all s ∈ R ;

( f 1 ) ′ ∫ V F ( x , 0 ) d μ = 0 and there exists a x 0 ∈ V such that F ( x 0 , 0 ) = 0 ;

( f 2 ) ′ there exist a constant α ∈ [ 0 , p ) and functions g i : V → R , i = 1 , 2 , with g 1 ∈ L ∞ ( V ) and g 2 ∈ L 1 ( V ) , such that

F ( x , s ) ≤ g 1 ( x ) ∣ s ∣ α + g 2 ( x )

for all ( x , s ) ∈ V × R ;

( f 3 ) ′ there are positive constants γ and δ with δ > γ κ , such that

Θ 1 ′ ≔ 1 γ p max ∣ s ∣ ≤ 1 h 0 1 ⁄ q μ 0 1 ⁄ q ( p γ p ) 1 p a ( ∣ s ∣ ) ∫ V b ( x ) d μ < F ( x , δ ) d μ δ p p M ≔ Θ 2 ′ ,

where κ = 1 p ∫ V h ( x ) d μ − p and

M = deg ( x 0 ) 2 μ ( x 0 ) p 2 μ ( x 0 ) + h ( x 0 ) μ ( x 0 ) + ∑ y ∼ x 0 w x 0 y 2 μ ( y ) p 2 μ ( y ) .

Then, for each parameter λ belonging to ( Θ 2 ′ − 1 , Θ 1 ′ − 1 ) , equation (5.2) has at least three solutions.

Remark 5.3

In Theorems 5.1 and 5.2, all three solutions are nontrivial solutions if we furthermore assume that f ( x , 0 ) ≠ 0 for some x ∈ V .

6 Examples

In this section, we present two examples as applications of Theorems 1.1 and 5.2.

Example 6.1

Let p = 2 , q = 3 , and m = 2 in (1.14). Consider the following system:

(6.1) £ 2 , 2 u + h 1 ( x ) u = λ F u ( x , u , v ) , x ∈ V , £ 2 , 3 v + h 2 ( x ) v = λ F v ( x , u , v ) , x ∈ V ,

where G = ( V , E ) is a finite graph of 9 vertexes, i.e., V = { x 1 , x 2 , … , x 9 } , the measure μ ( x i ) = 1 , i = 1 , 2 , … , 9 , h i : V → R + with h i ≡ 9 , i = 1 , 2 , for all x ∈ V and put

ω 1 = ( p γ 1 p + p γ 2 q ) 1 p h 1 , min 1 ⁄ p μ min 1 ⁄ p , ω 2 = ( q γ 1 p + q γ 2 q ) 1 q h 2 , min 1 ⁄ q μ min 1 ⁄ q .

Let λ > 0 and F : V × R × R → R is defined by

∂ F ( x , s , t ) ∂ s = ω 1 − ∣ s ∣ , 0 ≤ ∣ s ∣ ≤ ω 1 , ∣ s ∣ 3 − ω 1 3 , ω 1 < ∣ s ∣ < 4 ω 1 , ( 4 ω 1 ) r 1 ∣ s ∣ 3 − r 1 − ω 1 3 , 4 ω 1 ≤ ∣ s ∣ ,

and

∂ F ( x , s , t ) ∂ t = ω 2 − ∣ t ∣ , 0 ≤ ∣ t ∣ ≤ ω 2 , ∣ t ∣ 4 − ω 2 4 , ω 2 < ∣ t ∣ < 5 ω 2 , ( 5 ω 2 ) r 2 ∣ t ∣ 4 − r 2 − ω 2 4 , 5 ω 2 ≤ ∣ t ∣ .

Then,

F ( x , s , t ) = 1 2 ω 1 2 + 1 4 ( 4 ω 1 ) 4 + 3 4 ω 1 4 + 1 4 − r 1 ( 4 ω 1 ) r 1 ∣ s ∣ 4 − r 1 − ω 1 3 ∣ s ∣ − 1 4 − r 1 ( 4 ω 1 ) 4 + 1 2 ω 2 2 + 1 5 ( 5 ω 2 ) 5 + 4 5 ω 2 5 + 1 5 − r 2 ( 5 ω 2 ) r 2 ∣ t ∣ 5 − r 2 − ω 2 4 ∣ t ∣ − 1 5 − r 2 ( 5 ω 2 ) 5 , 4 ω 1 ≤ ∣ s ∣ , 5 ω 2 ≤ ∣ t ∣ , ω 1 ∣ s ∣ − 1 2 ∣ s ∣ 2 + 1 2 ω 2 2 + 1 5 ∣ t ∣ 5 − ω 2 4 ∣ t ∣ + 4 5 ω 2 5 , 0 ≤ ∣ s ∣ ≤ ω 1 , ω 2 < ∣ t ∣ < 5 ω 2 , ω 1 ∣ s ∣ − 1 2 ∣ s ∣ 2 + 1 2 ω 2 2 + 1 5 ( 5 ω 2 ) 5 + 4 5 ω 2 5 + 1 5 − r 2 ( 5 ω 2 ) r 2 ∣ t ∣ 5 − r 2 − ω 2 4 ∣ t ∣ − 1 5 − r 2 ( 5 ω 2 ) 5 , 0 ≤ ∣ s ∣ ≤ ω 1 , 5 ω 2 ≤ ∣ t ∣ , 1 2 ω 1 2 + 1 4 ∣ s ∣ 4 − ω 1 3 ∣ s ∣ + 3 4 ω 1 4 + ω 2 ∣ t ∣ − 1 2 ∣ t ∣ 2 , ω 1 < ∣ s ∣ < 4 ω 1 , 0 ≤ ∣ t ∣ ≤ ω 2 , 1 2 ω 1 2 + 1 4 ∣ s ∣ 4 − ω 1 3 ∣ s ∣ + 3 4 ω 1 4 + 1 2 ω 2 2 + 1 5 ∣ t ∣ 5 − ω 2 4 ∣ t ∣ + 4 5 ω 2 5 , ω 1 < ∣ s ∣ < 4 ω 1 , ω 2 < ∣ t ∣ < 5 ω 2 , 1 2 ω 1 2 + 1 4 ∣ s ∣ 4 − ω 1 3 ∣ s ∣ + 3 4 ω 1 4 + 1 2 ω 2 2 + 1 5 ( 5 ω 2 ) 5 + 4 5 ω 2 5 + 1 5 − r 2 ( 5 ω 2 ) r 2 ∣ t ∣ 5 − r 2 − ω 2 4 ∣ t ∣ − 1 5 − r 2 ( 5 ω 2 ) 5 , ω 1 < ∣ s ∣ < 4 ω 1 , 5 ω 2 ≤ ∣ t ∣ , 1 2 ω 1 2 + 1 4 ( 4 ω 1 ) 4 + 3 4 ω 1 4 + 1 4 − r 1 ( 4 ω 1 ) r 1 ∣ s ∣ 4 − r 1 − ω 1 3 ∣ s ∣ − 1 4 − r 1 ( 4 ω 1 ) 4 + ω 2 ∣ t ∣ − 1 2 ∣ t ∣ 2 , 4 ω 1 ≤ ∣ s ∣ , 0 ≤ ∣ t ∣ ≤ ω 2 , 1 2 ω 1 2 + 1 4 ( 4 ω 1 ) 4 + 3 4 ω 1 4 + 1 4 − r 1 ( 4 ω 1 ) r 1 ∣ s ∣ 4 − r 1 − ω 1 3 ∣ s ∣ − 1 4 − r 1 ( 4 ω 1 ) 4 + 1 2 ω 2 2 + 1 5 ∣ t ∣ 5 − ω 2 4 ∣ t ∣ + 4 5 ω 2 5 , 4 ω 1 ≤ ∣ s ∣ , ω 2 < ∣ t ∣ < 5 ω 2 , ω 1 ∣ s ∣ − 1 2 ∣ s ∣ 2 + ω 2 ∣ t ∣ − 1 2 ∣ t ∣ 2 , 0 ≤ ∣ s ∣ ≤ ω 1 , 0 ≤ ∣ t ∣ ≤ ω 2 ,

where ( r 1 , r 2 ) ∈ ( 1 , 2 ] × ( 1 , 3 ] . Next we verify that h 1 , h 2 , and F satisfy the conditions in Theorem 1.1.

  • Obviously, h i satisfy ( H ) , i = 1 , 2 , and F satisfies ( F 0 ) and ( F 1 ) .

  • Let

    f 1 ( x ) ≡ ( 4 ω 1 ) r 1 4 − r 1 , f 2 ( x ) ≡ ( 5 ω 2 ) r 2 5 − r 2

    and

    g ( x ) ≡ 1 2 ω 1 2 + 1 4 ( 4 ω 1 ) 4 + 3 4 ω 1 4 + 1 2 ω 2 2 + 1 5 ( 5 ω 2 ) 5 + 4 5 ω 2 5 , for all x ∈ V .

    Then,

    F ( x , s , t ) ≤ f 1 ( x ) ∣ s ∣ α + f 2 ( x ) ∣ t ∣ β + g ( x ) ,

    where α ∈ [ 0 , p ) , β ∈ [ 0 , q ) . Hence, F satisfies ( F 2 ) .

  • Let

    δ 1 = 4 ω 1 = 4 × 1 5 1 2 , δ 2 = 5 ω 2 = 5 × 45 2 1 3 , γ 1 = 81 2 1 2 , γ 2 = 81 3 1 3 .

    Then, δ 1 > γ 1 κ 1 = 1 , δ 2 > γ 2 κ 2 = 1 ,

    Λ 2 − 1 = δ 1 p p ∫ V h 1 ( x ) d μ + δ 2 q q ∫ V h 2 ( x ) d μ inf x ∈ V F ( x , δ 1 , δ 2 ) ∣ V ∣ = 81 2 ⋅ ( 4 ⋅ 1 5 1 2 ) 2 + 81 3 ⋅ ( 5 ⋅ 45 2 1 3 ) 3 1 2 ⋅ 1 5 2 2 + 1 4 ( 4 ⋅ 1 5 1 2 ) 4 − 4 ⋅ 1 5 4 2 + 3 4 ⋅ 1 5 4 2 + 1 2 ⋅ 45 2 2 3 + 1 5 ( 5 ⋅ 45 2 1 3 ) 5 − 5 ⋅ 45 2 5 3 + 4 5 ⋅ 45 2 5 3 × 9 ≈ 0.07614

    and

    Λ 1 − 1 = γ 1 p + γ 2 q max x ∈ V , ∣ s ∣ ≤ ( p γ 1 p + p γ 2 q ) 1 p h 1 , min 1 ⁄ p μ min 1 ⁄ p , ∣ t ∣ ≤ ( q γ 1 p + q γ 2 q ) 1 q h 2 , min 1 ⁄ q μ min 1 ⁄ q F ( x , s , t ) ∣ V ∣ = 81 2 + 81 3 1 2 ⋅ ( 15 ) 2 2 + 1 2 ⋅ 45 2 2 3 × 9 ≈ 0.65303 > Λ 2 − 1 .

    Hence, ( F 3 ) holds. Thus, by Theorem 1.1, for each λ ∈ ( Λ 2 − 1 , Λ 1 − 1 ) ≈ ( 0.07614 , 0.65303 ) , system (6.1) has at least three distinct solutions.

Example 6.2

Let p = 3 in (5.2). Consider the following scalar equation on locally finite graph G = ( V , E ) :

(6.2) − Δ 3 u + h ( x ) u = λ f ( x , u ) , x ∈ V ,

where the measure μ ( x ) ≡ 1 and h ( x ) ≡ 4 for all x ∈ V . For some fixed x 0 ∈ V , there are four edges x 0 y ∈ E with w x 0 y = 2 . Put

ω = ( p γ p ) 1 p h 0 1 ⁄ p μ 0 1 ⁄ p .

Let λ > 0 and F : V × R → R is defined by

f ( x 0 , s ) = ∂ F ( x 0 , s ) ∂ s = ω − ∣ s ∣ , ∣ s ∣ ≤ ω , ∣ s ∣ 5 − ω 5 , ω < ∣ s ∣ ≤ 6 ω , ( 6 ω ) r s 5 − r − ω 5 , 6 ω < ∣ s ∣ ,

and

f ( x , s ) = 0 , for all x ∈ V ⁄ { x 0 } .

Then,

F ( x 0 , s ) = ω ∣ s ∣ − 1 2 ∣ s ∣ 2 , ∣ s ∣ ≤ ω , 1 2 ω 2 + 1 6 ∣ s ∣ 6 − ω 5 ∣ s ∣ + 5 6 ω 6 , ω < ∣ s ∣ ≤ 6 ω , 1 2 ω 2 + 6 6 + 5 6 ω 6 − 1 6 − r ( 6 ω ) 6 + 1 6 − r ( 6 ω ) r ∣ s ∣ 6 − r − ω 5 ∣ s ∣ , 6 ω < ∣ s ∣

and

F ( x , s ) = 0 , for all x ∈ V ⁄ { x 0 } ,

where r ∈ ( 3 , 5 ] and λ > 0 . Next we verify that h and F satisfy the conditions in Theorem 5.2.

  • Obviously, h satisfies ( h ) ′ .

  • Let

    g 1 ( x ) = 1 6 − r ( 6 ω ) r , x = x 0 , 0 , x ≠ x 0 ,

    g 2 ( x ) = 1 2 ω 2 + 6 6 + 5 6 ω 6 , x = x 0 , 0 , x ≠ x 0 ,

    a ( ∣ s ∣ ) = ω ∣ s ∣ − 1 2 ∣ s ∣ 2 + 1 , ∣ s ∣ ≤ ω , 1 2 ω 2 + 1 6 ∣ s ∣ 6 − ω 5 ∣ s ∣ + 5 6 ω 6 + 1 , ω < ∣ s ∣ ≤ 6 ω , 1 2 ω 2 + 6 6 + 5 6 ω 6 − 1 6 − r ( 6 ω ) 6 + 1 6 − r ( 6 ω ) r ∣ s ∣ 6 − r − ω 5 ∣ s ∣ + 1 , 6 ω < ∣ s ∣ ,

    and

    b ( x ) = 1 , x = x 0 , 0 , x ≠ x 0 .

    Then,

    ‖ g 1 ‖ ∞ = 1 6 − r ( 6 ω ) r , ‖ g 2 ‖ L 1 ( V ) = 1 2 ω 2 + 6 6 + 5 6 ω 6

    and

    F ( x , s ) ≤ g 1 ( x ) ∣ s ∣ α + g 2 ( x ) .

    Moreover,

    f ( x , s ) ≤ a ( ∣ s ∣ ) b ( x ) , F ( x , s ) ≤ a ( ∣ s ∣ ) b ( x ) ,

    for all x ∈ V and all s ∈ R . Hence, F satisfies ( f 0 ) ′ , ( f 1 ) ′ , and ( f 2 ) ′ .

  • Let

    δ = 6 ω = 6 ⋅ 4 1 3 , γ = 16 3 1 3 .

    Then, δ > γ κ = 1 ,

    Θ 2 − 1 = δ p p M inf x ∈ V F ( x , δ ) = ( 6 ⋅ 4 1 3 ) 3 × 1 3 × 16 1 2 ⋅ 4 2 3 + 6 5 ⋅ 4 2 − 6 ⋅ 4 2 + 5 6 ⋅ 4 2 ≈ 0.0371

    and

    Θ 1 − 1 = γ p max x ∈ V , ∣ s ∣ ≤ ( p γ p ) 1 p h 1 , min 1 ⁄ p μ min 1 ⁄ p a ( ∣ s ∣ ) ∫ V b ( x ) d μ = 16 3 1 2 ⋅ 4 2 3 + 1 ≈ 2.36 > Θ 2 − 1 .

    Hence, F satisfies ( f 3 ) ′ . Thus, by Theorem 5.2, for each λ ∈ ( Θ 1 − 2 , Θ 1 − 1 ) ≈ ( 0.0371 , 2.36 ) , equation (6.2) has at least three distinct solutions.

Acknowledgments

The authors sincerely thank the reviewers for their valuable comments.

  1. Funding information: This project was supported by Yunnan Fundamental Research Projects (Grant No: 202301AT070465) and Xingdian Talent Support Program for Young Talents of Yunnan Province.

  2. Author contributions: Yan Pang and Xingyong Zhang contributed to the main manuscript equally.

  3. Conflict of interest: The authors declare that they have no conflict of interest.

  4. Ethical approval: Not applicable.

  5. Data availability statement: No data were used in this study.

References

[1] S. Y. Chung and C. A. Berenstein, ω-harmonic functions and inverse conductivity problems on networks, SIAM J. Appl. Math. 65 (2005), 1200–1226, DOI: https://doi.org/10.1137/S0036139903432743. 10.1137/S0036139903432743Search in Google Scholar

[2] A. Grigor’yan, Y. Lin, and Y. Yang, Yamabe type equations on graphs, J. Differential Equations 261 (2016), no. 9, 4924–4943, DOI: https://doi.org/10.1016/j.jde.2016.07.011. 10.1016/j.jde.2016.07.011Search in Google Scholar

[3] A. Grigor’yan, Y. Lin, and Y. Yang, Existence of positive solutions to some nonlinear equations on locally finite graphs, Sci. China Math. 60 (2017), 1311–1324, DOI: https://doi.org/10.1007/s11425-016-0422-y. 10.1007/s11425-016-0422-ySearch in Google Scholar

[4] M. Imbesi, G. Molica Bisci, and D. D. Repovs, Elliptic problems on weighted locally finite graphs, Topol. Methods Nonlinear Anal. 61 (2023), no. 1, 501–526.Search in Google Scholar

[5] A. Pinamonti and G. Stefani, Existence and uniqueness theorems for some semi-linear equations on locally finite graphs, Proc. Amer. Math. Soc. 150 (2022), no. 11, 4757–4770, DOI: https://doi.org/10.1090/proc/16046. 10.1090/proc/16046Search in Google Scholar

[6] X. Yu, X. Zhang, J. Xie, and X. Zhang, Existence of nontrivial solutions for a class of poly-Laplacian system with mixed nonlinearity on graphs, Math. Methods Appl. Sci. 47 (2023), no. 4, 1750–1763, DOI: https://doi.org/10.1002/mma.9621. 10.1002/mma.9621Search in Google Scholar

[7] P. Yang and X. Zhang, Existence and multiplicity of nontrivial solutions for a (p,q)-Laplacian system on locally finite graphs, Taiwanese J. Math. 28 (2024), no. 3, 551–588, DOI: https://doi.org/10.11650/tjm/240201. 10.11650/tjm/240201Search in Google Scholar

[8] J. Simon, Regularite de la solution d’une equation non lineaire dans Rn. In: Bénilan, P., Robert, J. (Eds.), Journées d’Analyse Non Linéaire. Lecture Notes in Mathematics, vol. 665. Springer, Berlin, Heidelberg.Search in Google Scholar

[9] B. Ricceri, On a classical existence theorem for nonlinear elliptic equations, Experimental, Constructive and Nonlinear Analysis (M. Théra, ed.), CMS Conf. Proc., vol. 27, 2000, pp. 275–278. Search in Google Scholar

[10] X. Zhang, X. Zhang, J. Xie, and X. Yu, Existence and multiplicity of nontrivial solutions for poly-Laplacian systems on finite graphs, Bound. Value Probl. 2022 (2022), no. 32, 1–13, DOI: https://doi.org/10.1186/s13661-022-01613-1. 10.1186/s13661-022-01613-1Search in Google Scholar

[11] G. Bonanno and S. A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition, Appl. Anal. 89 (2010), no. 1, 1–10, DOI: https://doi.org/10.1080/00036810903397438. 10.1080/00036810903397438Search in Google Scholar

[12] G. Bonanno and G. M Bisci, Three weak solutions for elliptic Dirichlet problems, J. Math. Anal. Appl. 382 (2011), no. 1, 1–8, DOI: https://doi.org/10.1016/j.jmaa.2011.04.026. 10.1016/j.jmaa.2011.04.026Search in Google Scholar

[13] E. Zeidler, Nonlinear Functional Analysis and Its Applications, II/B, Springer-Verlag, New York, 1990.10.1007/978-1-4612-0981-2Search in Google Scholar
Received: 2023-10-09
Revised: 2024-04-19
Accepted: 2024-07-04
Published Online: 2024-11-18

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/dema-2024-0062
Keywords for this article
critical point theorem; ploy-Laplacian system; finite graph; locally finite graph; nontrivial solution
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