Homoclinic solutions in periodic partial difference equations

Peng Mei; Zhan Zhou; Jianshe Yu

doi:10.1515/anona-2025-0072

Article Open Access

Homoclinic solutions in periodic partial difference equations

Peng Mei , Zhan Zhou and Jianshe Yu

Published/Copyright: April 16, 2025

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From the journal Advances in Nonlinear Analysis Volume 14 Issue 1

Abstract

By using critical point theory in combination with periodic approximations, we obtain novel sufficient conditions for the existence of nontrivial homoclinic solutions for a class of periodic partial difference equations with sign-changing mixed nonlinearities. Notably, in certain specific cases, a necessary and sufficient condition is also established.

Keywords: homoclinic solution; partial difference equation; discrete nonlinear Schrödinger equation; critical point theory; periodic approximation

MSC 2010: Primary: 39A14

1 Introduction

In this article, we consider the periodic partial difference equation:

(1.1) − Δ 1 2 u ( m − 1 , n ) − Δ 2 2 u ( m , n − 1 ) + ω ( m , n ) u ( m , n ) = f ( m , n , u ( m , n ) ) , ( m , n ) ∈ Z 2 ,

where Δ 1 u ( m , n ) = u ( m + 1 , n ) − u ( m , n ) , Δ 2 u ( m , n ) = u ( m , n + 1 ) − u ( m , n ) , Δ 1 2 u ( m , n ) = Δ 1 ( Δ 1 u ( m , n ) ) , and ω ( m + T 1 , n ) = ω ( m , n ) = ω ( m , n + T 2 ) > 0 for given positive integers T 1 , T 2 , f ( m , n , u ) is continuous in u ∈ R , and f ( m + T 1 , n , u ) = f ( m , n , u ) = f ( m , n + T 2 , u ) for each ( m , n ) ∈ Z 2 .

The problem arises when we seek standing waves of the more general two-dimensional (2D) periodic discrete nonlinear Schrödinger (DNLS) equation:

i ψ ˙ m , n = − Ξ ψ m , n + ε m , n ψ m , n − f ( m , n , ψ m , n ) , ( m , n ) ∈ Z 2 ,

where f ( m , n , ⋅ ) ∈ C ( R , R ) for each ( m , n ) ∈ Z 2 , and the nonlinearity is gauge invariant, i.e.,

f ( m , n , e i θ u ) = e i θ f ( m , n , u ) , θ ∈ R .

Making use of the standing wave ansata

ψ m , n = u m , n e − i ω t , lim ∣ m ∣ + ∣ n ∣ → ∞ ψ m , n = 0 ,

where { u m , n } is a real number sequence and ω ∈ R is the temporal frequency, we arrive at the partial difference equation (1.1) and

(1.2) lim ∣ m ∣ + ∣ n ∣ → ∞ u ( m , n ) = 0 ,

where ω ( m , n ) = ε m , n − ω for each ( m , n ) ∈ Z 2 .

We assume that f ( m , n , 0 ) = 0 for each ( m , n ) ∈ Z 2 , then { u ( m , n ) } = { 0 } is a solution of (1.1), which is called the trivial solution. As usual, we say that a solution u = { u ( m , n ) } of (1.1) is homoclinic (to 0) if (1.2) holds. In addition, if { u ( m , n ) } ≠ { 0 } , then u is called a nontrivial homoclinic solution. We are interested in the existence of nontrivial homoclinic solutions for (1.1).

The DNLS equation has naturally received special attention as a classical nonlinear lattice model and occurs in a wide variety of systems, such as the DNA double helix [23], nonlinear optics [4], complex electronic materials [26], and Bose-Einstein condensates [18]. More details on DNLS equations can be found in reviews [6,13]. At the same time, the 2D system plays an important role. As an example, consider the following 2D DNLS equation

(1.3) i ψ ˙ m , n + C Ξ ψ m , n + ∣ ψ m , n ∣ 2 ψ m , n = 0 , ( m , n ) ∈ Z 2 ,

where C = 1 ⁄ h 2 is the coupling constant ( h is the lattice spacing), and Ξ is the 2D discrete Laplacian operator defined by Ξ ψ m , n = ψ m , n + 1 + ψ m + 1 , n + ψ m , n − 1 + ψ m − 1 , n − 4 ψ m , n . It has been used to study semi-infinite 2D optical waveguide arrays with horizontal edges. More than this, there is another physical realization of the system. In fact, equation (1.3) describes the dynamical behaviour of a Bose-Einstein condensate trapped in a strong 2D optical lattice, where the periodic potential is induced by coherent laser beams [25].

In recent years, many researchers have carried out extensive and in-depth studies on the existence of homoclinic solutions for nonlinear discrete systems. To mention a few, see [1,15,21,28,29,31,32] and [9,14,16] for related topics. Breathers also exist in higher dimensions, as stated in the general theory of MacKay and Aubry [19]. Many of their basic properties appear in higher dimensions. In [8], Fleischer and his collaborators created a 2D photonic lattice using optical induction and reported experimental observations of 2D lattice solitons. Further theoretical and numerical simulation results are described in [7,27].

On the other hand, partial difference equations appeared early on, but unfortunately their further development did not continue until the late twentieth century [3]. Until now, however, not much research has been done on the qualitative theory of partial difference equations [2,5,11,17,22,30], except dealing with issues such as oscillation, stability, chaos, and boundary value problems, and there has been very little discussion of homoclinic solutions [12], mainly due to the lack of effective tools suitable for the study of partial difference equations. In fact, the critical point theory has become a very mature and useful research tool since it was introduced into difference equations to study periodic solutions [10]. We find that many models of partial difference equations, which are more relevant to practical applications than normal difference equations, have a variational structure. This allows us to try to use variational methods to study problems related to partial difference equations.

Based on these important meanings, it will be a topic of great theoretical significance and wide application prospects to study homoclinic solutions of partial difference equations by variational methods and to apply them to the study of properties of discrete solitons for multidimensional DNLS equations. Indeed, we have already made such an attempt and achieved a small breakthrough. In [20], a class of partial difference equations with mixed nonlinearities and unbounded potentials is studied. Using the critical point theory, some sufficient conditions for the existence and multiplicity of homoclinic solutions are obtained.

In this article, we attempt to investigate the existence of nontrivial homoclinic solutions of the periodic partial difference equation (1.1) using the critical point theory. To the best of our knowledge, such conditions are being presented for the first time within the domain of ordinary difference equations. It is worth noting that our conditions allow the limits of f ( m , n , u ) ⁄ u for each ( m , n ) ∈ Z 2 to be nonexistent at both the origin and infinity. When the limits exist, our conditions also contain superlinear, asymptotically linear, and a mixture of them. Moreover, we also allow the nonlinear terms to change sign, and there is no monotonic constraint on f ( m , n , u ) ⁄ u for ( m , n ) ∈ Z 2 . Even for ordinary difference equations, our results significantly improve some existing ones. We also give an example and the corresponding figure to show the superiority of our results. Finally, we present a corollary with monotonic conditions, which leads to a necessary and sufficient condition for the existence of nontrivial homoclinic solutions of equation (1.1). Details can be found in the remarks.

We denote by l 2 the set of all functions u : Z 2 → R such that

‖ u ‖ = ∑ m = − ∞ ∞ ∑ n = − ∞ ∞ u 2 ( m , n ) 1 2 < ∞ .

On the Hilbert space l 2 , we consider the functional

J ( u ) = ∑ m = − ∞ ∞ ∑ n = − ∞ ∞ 1 2 ( Δ 1 u ( m − 1 , n ) ) 2 + 1 2 ( Δ 2 u ( m , n − 1 ) ) 2 + 1 2 ω ( m , n ) u 2 ( m , n ) − F ( m , n , u ( m , n ) ) .

Then J ∈ C 1 ( l 2 , R ) and

(1.4) ⟨ J ′ ( u ) , v ⟩ = ∑ m = − ∞ ∞ ∑ n = − ∞ ∞ [ − Δ 1 2 u ( m − 1 , n ) − Δ 2 2 u ( m , n − 1 ) + ω ( m , n ) u ( m , n ) − f ( m , n , u ( m , n ) ) ] v ( m , n ) , u , v ∈ l 2 .

Equation (1.4) implies that (1.1) is the corresponding Euler-Lagrange equation for J . To find nontrivial homoclinic solutions of (1.1), we need only to look for nonzero critical points of J in l 2 .

Then, we establish the main results in this section:

Theorem 1.1

Assume that f ( m , n , u ) is continuous in u , and f ( m + T 1 , n , u ) = f ( m , n , u ) = f ( m , n + T 2 , u ) for each ( m , n ) ∈ Z 2 and u ∈ R . For ( m , n ) ∈ Z 2 , the following conditions hold.

limsup u → 0 f ( m , n , u ) u = a ( m , n ) < ω ( m , n ) and liminf u → 0 f ( m , n , u ) u = b ( m , n ) > − ω ( m , n ) .
liminf ∣ u ∣ → ∞ F ( m , n , u ) u 2 = c ( m , n ) ≤ ∞ , where F ( m , n , u ) = ∫ 0 u f ( m , n , s ) d s .
f ( m , n , u ) u − 2 F ( m , n , u ) > 0 for u ≠ 0 and f ( m , n , u ) u − 2 F ( m , n , u ) → ∞ as ∣ u ∣ → ∞ .

If 2 c ( m , n ) > ω ( m , n ) for each ( m , n ) ∈ Z 2 , then (1.1)has at least one nontrivial solution in l 2 .

Remark 1.1

The conditions ( F 1 ) and ( F 2 ) allow for the nonexistence of limits of f ( m , n , u ) ⁄ u for each ( m , n ) ∈ Z 2 both at the origin and at infinity, which of course means that our conditions encompass cases of superlinear, asymptotically linear, and a mixture of them. In comparison with similar results for ordinary difference equations [1,21,29,31,32], we do not have to require that f is only superlinear or asymptotically linear at the origin or at infinity.

Remark 1.2

We allow nonlinear terms to change sign and do not require nonnegativity as in other articles [21,31,32].

Remark 1.3

In comparison with the conditions in [32], we remove the following condition:

limsup u → 0 f n 2 ( u ) f n ( u ) u − 2 F n ( u ) = p n < ∞ .

Remark 1.4

We introduce another condition in [32] that f ( m , n , u ) ⁄ u is strictly increasing in ( 0 , ∞ ) and strictly decreasing in ( − ∞ , 0 ) for ( m , n ) ∈ Z 2 , which implies that f ( m , n , u ) u − 2 F ( m , n , u ) > 0 for u ≠ 0 . In fact, let

f ( m , n , u ) = τ ( m , n ) 6 u 3 1 + u 2 , 0 < ∣ u ∣ < 1 , τ ( m , n ) ( ∣ u ∣ 3 − 6 u 2 + 9 ∣ u ∣ − 1 ) u , 1 ≤ ∣ u ∣ ≤ 6 5 , τ ( m , n ) 22,021 u 3 4,500 ( 1 + u 2 ) , 6 5 < ∣ u ∣ ,

where τ ( m + T 1 , n ) = τ ( m , n ) = τ ( m , n + T 2 ) are positive real numbers. As shown in Figure 1, it is easy to see that f ( m , n , u ) ⁄ u is not strictly increasing in ( 0 , ∞ ) and not strictly decreasing in ( − ∞ , 0 ) for ( m , n ) ∈ Z 2 .

Nevertheless, f can satisfy all conditions in Theorem 1.1, implying that we have no monotone constraint on f ( m , n , u ) ⁄ u for ( m , n ) ∈ Z 2 .

$Figure 1 The images of f ( m , n , u ) ⁄ u f\left(m,n,u)/u for τ ( m , n ) ≡ 1 \tau \left(m,n)\equiv 1 .$

Figure 1

The images of f ( m , n , u ) ⁄ u for τ ( m , n ) ≡ 1 .

The following corollary follows directly from Remark 1.4.

Corollary 1.1

lim u → 0 f ( m , n , u ) u = 0 .
lim ∣ u ∣ → ∞ F ( m , n , u ) u 2 = c ¯ ( m , n ) < ∞ , where F ( m , n , u ) = ∫ 0 u f ( m , n , s ) d s .
f ( m , n , u ) ⁄ u is strictly increasing in ( 0 , ∞ ) and strictly decreasing in ( − ∞ , 0 ) .
f ( m , n , u ) u − 2 F ( m , n , u ) → ∞ as ∣ u ∣ → ∞ .

If 2 c ¯ ( m , n ) > ω ( m , n ) for each ( m , n ) ∈ Z 2 , then (1.1) has at least one nontrivial solution in l 2 .

Further, if some of the conditions are not satisfied, then (1.1) has no nontrivial solution in l 2 , as shown in the following proposition.

Proposition 1.1

Suppose that the conditions of Corollary 1.1 hold. If 2 c ¯ ( m , n ) ≤ ω ( m , n ) for each ( m , n ) ∈ Z 2 , then (1.1) has no nontrivial solution in l 2 .

Proof of Proposition 1.1

By way of contradiction, we assume that (1.1) has a nontrivial solution u = { u ( m , n ) } ∈ l 2 . Then u is a nonzero critical point of J , and

⟨ J ′ ( u ) , u ⟩ = ∑ m = − ∞ ∞ ∑ n = − ∞ ∞ [ ( Δ 1 u ( m − 1 , n ) ) 2 + ( Δ 2 u ( m , n − 1 ) ) 2 + ω ( m , n ) u 2 ( m , n ) − f ( m , n , u ( m , n ) ) u ( m , n ) ] = 0 .

Then, by ( G 2 ) and ( G 3 ) , we have

∑ m = − ∞ ∞ ∑ n = − ∞ ∞ ω ( m , n ) u 2 ( m , n ) ≤ ∑ m = − ∞ ∞ ∑ n = − ∞ ∞ f ( m , n , u ( m , n ) ) u ( m , n ) < ∑ m = − ∞ ∞ ∑ n = − ∞ ∞ 2 c ¯ ( m , n ) u 2 ( m , n ) .

This is impossible as 2 c ¯ ( m , n ) ≤ ω ( m , n ) for ( m , n ) ∈ Z 2 and the proof is complete.□

Necessary and sufficient condition 1

Combining Corollary 1.1 and Proposition 1.1 gives the necessary and sufficient condition for the existence of nontrivial solutions to equation (1.1); that is, if the conditions of Corollary 1.1 hold, and c ¯ ( m , n ) ≡ c , ω ( m , n ) ≡ ω for ( m , n ) ∈ Z 2 , then (1.1) has at least one nontrivial solution in l 2 if and only if 2 c > ω .

The remaining of this article is organized as follows. In Section 2, we establish the variational framework associated with (1.1) and cite the Mountain Pass Lemma. We then give some crucial lemmas and complete our proof of Theorem 1.1 in Section 3.

2 The variational structure

We first establish the variational setting associated with (1.1). Let S be the vector space of all real sequences of the form

u = { u ( m , n ) } ( m , n ) ∈ Z 2 = { u ( 0 , 0 ) , u ( 1 , 0 ) , u ( 0 , 1 ) , u ( − 1 , 0 ) , u ( 0 , − 1 ) , u ( 2 , 0 ) , u ( 1 , 1 ) , u ( 0 , 2 ) , … , u ( m , n ) , … } ,

namely,

S = { u = { u ( m , n ) } ∣ u ( m , n ) ∈ R , ( m , n ) ∈ Z 2 } .

Then S is a vector space with a u + b v = { a u ( m , n ) + b v ( m , n ) } for u , v ∈ S , a , b ∈ R . For any fixed positive integer k , we define the subspace E k as follows:

E k = { u ∈ S ∣ u ( m + 2 k T 1 , n ) = u ( m , n ) = u ( m , n + 2 k T 2 ) , ( m , n ) ∈ Z 2 } .

Obviously, E k can be equipped with the inner product ( ⋅ , ⋅ ) k and norm ‖ ⋅ ‖ k given by

( u , v ) k = ∑ m = − k T 1 k T 1 − 1 ∑ n = − k T 2 k T 2 − 1 u ( m , n ) v ( m , n ) , u , v ∈ E k ,

and

‖ u ‖ k = ∑ m = − k T 1 k T 1 − 1 ∑ n = − k T 2 k T 2 − 1 u 2 ( m , n ) 1 2 , u ∈ E k ,

respectively. We also define a norm ‖ ⋅ ‖ k ∞ by

‖ u ‖ k ∞ = max { ∣ u ( m , n ) ∣ : − k T 1 ≤ m ≤ k T 1 − 1 , − k T 2 ≤ n ≤ k T 2 − 1 } , u ∈ E k .

On E k , we consider the functional

(2.1) J k ( u ) = ∑ m = − k T 1 k T 1 − 1 ∑ n = − k T 2 k T 2 − 1 1 2 ( Δ 1 u ( m − 1 , n ) ) 2 + 1 2 ( Δ 2 u ( m , n − 1 ) ) 2 + 1 2 ω ( m , n ) u 2 ( m , n ) − F ( m , n , u ( m , n ) ) .

Standard arguments show that the functional J k satisfies

(2.2) ⟨ J k ′ ( u ) , v ⟩ = ∑ m = − k T 1 k T 1 − 1 ∑ n = − k T 2 k T 2 − 1 [ − Δ 1 2 u ( m − 1 , n ) − Δ 2 2 u ( m , n − 1 ) + ω ( m , n ) u ( m , n ) − f ( m , n , u ( m , n ) ) ] v ( m , n ) , u , v ∈ E k .

Since ω and f are periodic, it is straightforward to show that the critical points of J k in E k are exactly periodic solutions of system (1.1).

Next, we give some basic notations and some known results from critical point theory.

Let E be a real Banach space and C 1 ( E , R ) denote the set of functionals that are Fréchet differentiable, and their Fréchet derivatives are continuous on E .

Definition 2.1

Let J ∈ C 1 ( E , R ) . A sequence { x j } ⊂ E is called a Cerami sequence for J if J ( x j ) → c for some c ∈ R and ( 1 + ‖ x j ‖ ) J ′ ( x j ) → 0 as j → ∞ . We say J satisfies the Cerami condition if any Cerami sequence for J possesses a convergent subsequence.

Lemma 2.1

(Mountain Pass Lemma [24]) Let J ∈ C 1 ( E , R ) satisfy the following conditions: there exist e ∈ E \ { 0 } and r ∈ ( 0 , ‖ e ‖ ) such that max { J ( 0 ) , J ( e ) } < inf ‖ u ‖ = r J ( u ) . Then there exists a Cerami sequence { u n } for the mountain pass level c which is defined by

c = inf h ∈ Γ max s ∈ [ 0 , 1 ] J ( h ( s ) ) ,

where

Γ = { h ∈ C ( [ 0 , 1 ] , E ) : h ( 0 ) = 0 , h ( 1 ) = e } .

3 Proofs of main results

Before proving the main results of this article, we need to establish some lemmas.

Lemma 3.1

Under the assumptions of Theorem 1.1, the functional J k satisfies the Cerami condition.

Proof

Let { u j } ⊂ E k be a Cerami sequence for J k , that is,

J k ( u j ) → c , ( 1 + ‖ u j ‖ k ) J k ′ ( u j ) → 0 , as j → ∞ .

We need to show that { u j } has a convergent subsequence. Thanks to E k being finite dimensional, it suffices to show that ‖ u j ‖ k is bounded, then { J k ( u j ) } and { ( 1 + ‖ u j ‖ k ) J k ′ ( u j ) } are bounded. Without loss of generality, we assume that ∣ J k ( u j ) ∣ ≤ M − 1 2 for M > 2 ∣ c ∣ + 1 and ‖ ( 1 + ‖ u j ‖ k ) J k ′ ( u j ) ‖ < 1 for j ∈ N . Then by (2.1) and (2.2), we have

(3.1) ∑ m = − k T 1 k T 1 − 1 ∑ n = − k T 2 k T 2 − 1 ( f ( m , n , u j ( m , n ) ) u j ( m , n ) − 2 F ( m , n , u j ( m , n ) ) ) = 2 J k ( u j ) − ⟨ J k ′ ( u j ) , u j ⟩ ≤ 2 ∣ J k ( u j ) ∣ + ‖ u j ‖ k ‖ J k ′ ( u j ) ‖ ≤ M .

From ( F 3 ) , there exists a positive constant ζ such that

f ( m , n , u ) u − 2 F ( m , n , u ) > M for ( m , n ) ∈ Z 2 and ∣ u ∣ > ζ ,

then (3.1) implies that ∣ u j ( m , n ) ∣ ≤ ζ for each ( m , n ) ∈ Z 2 , that is,

(3.2) ‖ u j ‖ k ∞ ≤ ζ .

Since E k is finite dimensional, it follows that ‖ ⋅ ‖ k and ‖ ⋅ ‖ k ∞ are equivalent. Then (3.2) implies that { ‖ u j ‖ k } is bounded. We have thus proved the lemma.□

Lemma 3.2

Under the assumptions of Theorem 1.1, there exists k 0 ∈ N such that J k has at least a nonzero critical point u k in E k for each k ≥ k 0 .

Proof

First, we need to show that J k satisfies conditions in Lemma 2.1. For any ε > 0 , set

p ( m , n ) = max { ∣ a ( m , n ) + ε ∣ , ∣ b ( m , n ) − ε ∣ } for ( m , n ) ∈ Z 2 .

By ( F 1 ) , there exists r > 0 such that

∣ f ( m , n , u ) ∣ ≤ p ( m , n ) ∣ u ∣ for ( m , n ) ∈ Z 2 and ∣ u ∣ ≤ r ,

and

∣ F ( m , n , u ) ∣ ≤ 1 2 p ( m , n ) u 2 for ( m , n ) ∈ Z 2 and ∣ u ∣ ≤ r .

Choosing

ε = 1 2 min ( m , n ) ∈ Z 2 { ω ( m , n ) − a ( m , n ) , ω ( m , n ) + b ( m , n ) } ,

we have

ω ( m , n ) ≥ p ( m , n ) + ε for ( m , n ) ∈ Z 2 .

Then, for u ∈ E k with ‖ u ‖ k ∞ ≤ ‖ u ‖ k = r ,

J k ( u ) ≥ 1 2 ∑ m = − k T 1 k T 1 − 1 ∑ n = − k T 2 k T 2 − 1 ω ( m , n ) u 2 ( m , n ) − ∑ m = − k T 1 k T 1 − 1 ∑ n = − k T 2 k T 2 − 1 F ( m , n , u ( m , n ) ) ≥ 1 2 ∑ m = − k T 1 k T 1 − 1 ∑ n = − k T 2 k T 2 − 1 ω ( m , n ) u 2 ( m , n ) − 1 2 ∑ m = − k T 1 k T 1 − 1 ∑ n = − k T 2 k T 2 − 1 p ( m , n ) u 2 ( m , n ) ≥ ε 2 ‖ u ‖ k 2 .

Take a = ε r 2 ⁄ 2 , then inf ‖ u ‖ k = r J k ( u ) ≥ a .

On the other hand, due to 2 c ( m , n ) > ω ( m , n ) for each ( m , n ) ∈ Z 2 , there exists a positive constant ε = ( 2 c ( m , n ) − ω ( m , n ) ) ⁄ 10 < 1 . Then choose e = { e ( m , n ) } ∈ l 2 with ‖ e ‖ = 1 such that

∑ m = − ∞ + ∞ ∑ n = − ∞ + ∞ ( Δ 1 e ( m − 1 , n ) ) 2 < ε , ∑ m = − ∞ + ∞ ∑ n = − ∞ + ∞ ( Δ 2 e ( m , n − 1 ) ) 2 < ε .

Let k 0 be large enough such that

∑ m = − k 0 T 1 k 0 T 1 − 1 ∑ n = − k 0 T 2 k 0 T 2 − 1 e 2 ( m , n ) ≥ 1 2 .

Let Ω = { ( m , n ) ∈ Z 2 : − k 0 T 1 ≤ m ≤ k 0 T 1 − 1 , − k 0 T 2 ≤ n ≤ k 0 T 2 − 1 } , for k ≥ k 0 , and define e k ∈ E k by

e k ( m , n ) = e ( m , n ) , ( m , n ) ∈ Ω ; 0 , ( m , n ) ∉ Ω .

By ( F 2 ) , there exists a constant t 0 > 2 r such that

F ( m , n , t e ( m , n ) ) ≥ t 2 ( c ( m , n ) − ε ) e 2 ( m , n ) for ( m , n ) ∈ Ω and t ≥ t 0 .

Then, for t ≥ t 0 ,

J k ( t e k ) = ∑ m = − k T 1 k T 1 − 1 ∑ n = − k T 2 k T 2 − 1 t 2 2 ( Δ 1 e k ( m − 1 , n ) ) 2 + t 2 2 ( Δ 2 e k ( m , n − 1 ) ) 2 + t 2 2 ω ( m , n ) e k 2 ( m , n ) − F ( m , n , t e k ( m , n ) ) ≤ ε t 2 + ∑ m = − k 0 T 1 k 0 T 1 − 1 ∑ n = − k 0 T 2 k 0 T 2 − 1 t 2 2 ( ω ( m , n ) + 2 ε − 2 c ( m , n ) ) e 2 ( m , n ) ≤ − ε t 2 .

Thus,

J k ( t 0 e k ) ≤ − ε t 0 2 < 0 .

Thanks to ( F 1 ) , it follows that f ( m , n , 0 ) = 0 for each ( m , n ) ∈ Z 2 . Then we have r < ‖ t 0 e k ‖ k and

max { J k ( 0 ) , J k ( t 0 e k ) } = 0 < a ≤ inf ‖ u ‖ k = r J k ( u ) .

Now that we have verified all assumptions of Lemma 2.1. We know that J k possesses a critical value c k ≥ a with

c k = inf h ∈ Γ k max s ∈ [ 0 , 1 ] J k ( h ( s ) ) ,

where

Γ k = { h ∈ C ( [ 0 , 1 ] , E k ) : h ( 0 ) = 0 , h ( 1 ) = t 0 e k } .

A critical point u k of J k corresponding to c k is nonzero as c k ≥ a > 0 .□

Lemma 3.3

There exist two positive constants α and β such that

α ≤ ‖ u k ‖ k ∞ ≤ β

holds for every critical point u k of J k in E k with k ≥ k 0 , where k 0 is defined in Lemma 3.2.

Proof

For k ≥ k 0 , we define h k ∈ Γ k as h k ( s ) = s t 0 e k for s ∈ [ 0 , 1 ] , then

(3.3) J k ( u k ) ≤ max s ∈ [ 0 , 1 ] { J k ( s t 0 e k ) } ≤ max s ∈ [ 0 , 1 ] ∑ m = − k 0 T 1 k 0 T 1 − 1 ∑ n = − k 0 T 2 k 0 T 2 − 1 4 ∣ s t 0 e ( m , n ) ∣ 2 + 1 2 ω ( m , n ) ∣ s t 0 e ( m , n ) ∣ 2 − F ( m , n , s t 0 e ( m , n ) ) ≜ M 0 .

Obviously, M 0 > 0 is independent of k .

Since u k is a critical point of J k , by (2.1), (2.2), and (3.3), we have

(3.4) ∑ m = − k T 1 k T 1 − 1 ∑ n = − k T 2 k T 2 − 1 ( f ( m , n , u k ( m , n ) ) u k ( m , n ) − 2 F ( m , n , u k ( m , n ) ) ) ≤ 2 M 0 .

From ( F 3 ) , there exists a constant β > 0 such that

f ( m , n , u ) u − 2 F ( m , n , u ) > 2 M 0 for ( m , n ) ∈ Z 2 and ∣ u ∣ > β ,

then (3.4) implies that ∣ u k ( m , n ) ∣ ≤ β for each ( m , n ) ∈ Z 2 , that is,

‖ u k ‖ k ∞ ≤ β .

From (2.2), we have

(3.5) ∑ m = − k T 1 k T 1 − 1 ∑ n = − k T 2 k T 2 − 1 ω ( m , n ) u k 2 ( m , n ) ≤ ∑ m = − k T 1 k T 1 − 1 ∑ n = − k T 2 k T 2 − 1 f ( m , n , u k ( m , n ) ) u k ( m , n ) .

And there exists a constant α > 0 such that

(3.6) f ( m , n , u ) u ≤ p ( m , n ) u 2 for ( m , n ) ∈ Z 2 and ∣ u ∣ ≤ α ,

which together with (3.5) produces

‖ u k ‖ k ∞ ≥ α .

The proof is complete.□

Now, we are ready to prove Theorem 1.1.

Proof of Theorem 1.1

It follows from Lemma 3.2 that for each k > k 0 , J k has at least a nonzero critical point u k = { u k ( m , n ) } ∈ E k . By Lemma 3.3, there exists ( m k , n k ) ∈ Z 2 such that

(3.7) α ≤ ∣ u k ( m k , n k ) ∣ ≤ β .

Note that

(3.8) − Δ 1 2 u k ( m − 1 , n ) − Δ 2 2 u k ( m , n − 1 ) + ω ( m , n ) u k ( m , n ) = f ( m , n , u k ( m , n ) ) , ( m , n ) ∈ Z 2 .

By the periodicity of ω and f , we see that { u k ( m + T 1 , n + T 2 ) } is also a solution of (3.8). Making some shifts if necessary, without loss of generality, we can assume that 0 ≤ m k ≤ T 1 − 1 , 0 ≤ n k ≤ T 2 − 1 in (3.7). Moreover, passing to a subsequence of { u k } if necessary, we can also assume that ( m k , n k ) = ( m ∗ , n ∗ ) for k ≥ k 0 and some integer pair ( m ∗ , n ∗ ) such that 0 ≤ m ∗ ≤ T 1 − 1 , 0 ≤ n ∗ ≤ T 2 − 1 . It follows from (3.7) that we can choose a subsequence, still denoted by { u k } , such that

u k ( m , n ) → u ( m , n ) as k → ∞ , ( m , n ) ∈ Z 2 .

Then u = { u ( m , n ) } is a nonzero sequence as (3.7) implies ∣ u ( m ∗ , n ∗ ) ∣ ≥ α . It remains to show that u = { u ( m , n ) } ∈ l 2 , and that u is a solution of (1.1).

First, we show that u ∈ l 2 . Let

A k = { ( m , n ) ∈ Z 2 : ∣ u k ( m , n ) ∣ < α , − k T 1 ≤ m ≤ k T 1 − 1 , − k T 2 ≤ n ≤ k T 2 − 1 } , B k = { ( m , n ) ∈ Z 2 : ∣ u k ( m , n ) ∣ ≥ α , − k T 1 ≤ m ≤ k T 1 − 1 , − k T 2 ≤ n ≤ k T 2 − 1 } .

Since f ( m , n , u ) and F ( m , n , u ) are continuous in u and periodic in ( m , n ) , let

c 1 = max { f ( m , n , u ) u : α ≤ ∣ u ∣ ≤ β , ( m , n ) ∈ Z 2 } , c 2 = min { f ( m , n , u ) u − 2 F ( m , n , u ) : α ≤ ∣ u ∣ ≤ β , ( m , n ) ∈ Z 2 } .

From ( F 3 ) , it is clear that c 1 , c 2 > 0 . Then, by combining (3.5) and (3.6), we find that

∑ m = − k T 1 k T 1 − 1 ∑ n = − k T 2 k T 2 − 1 ω ( m , n ) u k 2 ( m , n ) ≤ ∑ m = − k T 1 k T 1 − 1 ∑ n = − k T 2 k T 2 − 1 f ( m , n , u k ( m , n ) ) u k ( m , n ) = ∑ ( m , n ) ∈ A k f ( m , n , u k ( m , n ) ) u k ( m , n ) + ∑ ( m , n ) ∈ B k f ( m , n , u k ( m , n ) ) u k ( m , n ) ≤ ∑ m = − k T 1 k T 1 − 1 ∑ n = − k T 2 k T 2 − 1 p ( m , n ) u k 2 ( m , n ) + c 1 c 2 ∑ ( m , n ) ∈ B k ( f ( m , n , u k ( m , n ) ) u k ( m , n ) − 2 F ( m , n , u k ( m , n ) ) ) ≤ ∑ m = − k T 1 k T 1 − 1 ∑ n = − k T 2 k T 2 − 1 p ( m , n ) u k 2 ( m , n ) + 2 c 1 M 0 c 2 ,

which implies that

(3.9) ‖ u k ‖ k 2 ≤ 2 c 1 M 0 c 2 ε .

For each κ ∈ N , let k > max { κ , k 0 } . Then it follows from (3.9) that

∑ m = − κ κ ∑ n = − κ κ u k 2 ( m , n ) ≤ ‖ u k ‖ k 2 ≤ 2 c 1 M 0 c 2 ε .

Letting k → ∞ gives us ∑ m = − κ κ ∑ n = − κ κ u 2 ( m , n ) ≤ 2 c 1 M 0 c 2 ε . By the arbitrariness of κ , we know that u = { u ( m , n ) } ∈ l 2 .

Finally, we show that u = { u ( m , n ) } satisfies (1.1). Indeed, for each ( m , n ) ∈ Z 2 , letting k → ∞ in (3.8) gives us follows:

− Δ 1 2 u ( m − 1 , n ) − Δ 2 2 u ( m , n − 1 ) + ω ( m , n ) u ( m , n ) = f ( m , n , u ( m , n ) ) , ( m , n ) ∈ Z 2 ,

that is, u = { u ( m , n ) } satisfies (1.1).

In summary, we show that u = { u ( m , n ) } is a nontrivial solution of (1.1) in l 2 . This completes the proof of Theorem 1.1.□

Acknowledgments

We would like to take this opportunity to thank the reviewers for their constructive and helpful comments and suggestions.

Funding information: This work is supported by the National Natural Science Foundation of China (Grant Nos. 12371184 and 12201141) and the Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT_16R16).
Author contributions: All authors contributed equally to the manuscript and read and approved the final manuscript.
Conflict of interest: The authors state no conflict of interest.

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Received: 2024-08-20

Revised: 2024-10-06

Accepted: 2025-01-23

Published Online: 2025-04-16

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

Articles in the same Issue

https://doi.org/10.1515/anona-2025-0072

Keywords for this article

homoclinic solution; partial difference equation; discrete nonlinear Schrödinger equation; critical point theory; periodic approximation

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