Three Nontrivial Solutions for Second-Order Partial Difference Equation via Morse Theory

Here, f ðði, jÞ, uÞ: Ω ×R⟶R is differentiable in u and there exists a function Fðði, jÞ, uÞ such that Fðði, jÞ, uÞ = Ð u0 f ðði, jÞ, τÞdτ for each ði, jÞ ∈Ω. Further, for all ði, jÞ ∈Ω, throughout this paper, we assume that ð f0Þf ðði, jÞ, 0Þ = 0, ðpÞpði, jÞ > 0, rði, jÞ > 0, qði, jÞ ≤ 0 and qði, jÞ≡0. As usual, if u satisfies (1.1)-(1.2), we say u is a solution of (1) and (2). According to ðf0Þ, it is easy to see that (1) and (2) admit a trivial solution u = 0. Meanwhile, what we care about is the existence and multiplicity of nontrivial solutions of (1)-(2). Consider (1), a partial difference equation, involving functions with two discrete variables, it can be used to many investigations related to image processing, population models, and digital control systems [1]. Due to the rapid development of modern digital computing devices, more and more important information about the behavior of complex systems can be revealed by simulations by modern digital computing devices in a simple way, which contributes greatly to the increasing interest in discrete problems and they are investigated in many literatures, for example, [2–8]. Let rði, j − 1Þ ≡ 0 in (1), it becomes an ordinary difference equation, namely,

As usual, if u satisfies (1.1)-(1.2), we say u is a solution of (1) and (2). According to ðf 0 Þ, it is easy to see that (1) and (2) admit a trivial solution u = 0. Meanwhile, what we care about is the existence and multiplicity of nontrivial solutions of (1)- (2).
Consider (1), a partial difference equation, involving functions with two discrete variables, it can be used to many investigations related to image processing, population models, and digital control systems [1]. Due to the rapid development of modern digital computing devices, more and more important information about the behavior of complex systems can be revealed by simulations by modern digital computing devices in a simple way, which contributes greatly to the increasing interest in discrete problems and they are investigated in many literatures, for example, [2][3][4][5][6][7][8].
Let rði, j − 1Þ ≡ 0 in (1), it becomes an ordinary difference equation, namely, (3) has captured many interests and has been studied extensively. Here, mention a few and [9] considered the existence of periodic solutions via critical theory. Ma and Guo [10] discussed homoclinic orbits. [11] got sign-changing solutions, whereas, when rði, j − 1Þ ≠ 0, it seems that there are rare literatures.
Moreover, (1) can be regarded as a discrete analog of a partial differential equation. It is well known that, with the rapid development of critical point theory, it becomes a more and more powerful to deal with the existence and multiplicity solutions of both partial differential equations and partial difference equations [12][13][14]. As mentioned, the Morse theory is a very useful tool to study the existence of multiple solutions of differential equations having variational structure, and it has been applied successfully to study differential equations [15][16][17][18]. At the same time, difference equations, regarded as discretizations of differential equations, are considered and multiple solutions are achieved via the Morse theory in some literatures [19,20]. However, there are few literatures using the Morse theory to study partial difference equations. Due to abovementioned reasons, we devote to studying the Dirichlet boundary value problem of second-order partial difference (equations (1) and (2)) by the Morse theory.
The organization of the rest of this paper reads as follows. In Section 2, we construct a suitable variational framework corresponding to (1) and (2) and reduce the existence of solutions of (1) and (2) to the existence of critical points of the associated functional. With preparation of Section 2, Section 3 not only displays the main result of this paper but also provides detailed proof of the main result. Finally, an example is exhibited to demonstrate our main result in Section 4.

Variational Structure and Some Auxiliary Results
In this section, we construct a variational functional corresponding to (1) and (2) on a suitable function space and state some basic facts. Let For any u, v ∈ S, define an inner product h·, · i by then the induced norm is Hence, ðS, h·, · iÞ is a T 1 T 2 -dimensional Hilbert space.
Consider the functional I : S ⟶ ℝ as the following form: Note that f ðði, jÞ, uÞ is differentiable in u, which ensures IðuÞ is twice differentiable. What is more, for any u, v ∈ S, make use of the boundary conditions (2), we have and we transfer the existence of nontrivial solutions of (1) and (2) into the existence of critical points of I on S.
In the following, we introduce some basic facts. Definition 1. [21]. The functional I satisfies the weaker Cerami condition (ðCÞ c condition for short) at the level c ∈ ℝ if any sequence fu n g ⊆ S satisfying Iðu n Þ ⟶ c, ð1 + ku n kÞkI ′ ðu n Þk ⟶ 0 as n ⟶ ∞ has a convergent subsequence. I satisfies ðCÞ condition if I satisfies ðCÞ c condition at any c ∈ ℝ.
Definition 2 [16,22]. Let u 0 be an isolated critical group of I with Iðu 0 Þ = c ∈ ℝ and U be a neighborhood of u 0 , the group is called the q-th critical group of I at u 0 . Let κ = fu ∈ S | I ′ ðuÞ = 0g. If IðκÞ is bounded from below by a ∈ ℝ and I satisfies ðDÞ c condition for all c ≤ a. Then, the group is called the q-th critical group of I at infinity.
In applications of Morse theory, it is necessary to make the functional satisfy the deformation condition ðDÞ, which is introduced by [23]. And [24] proves that once the functional I satisfies the ðCÞ condition, it must satisfy the deformation condition ðDÞ. Let S be a real Hilbert space and I ∈ C 2 ðS, ℝÞ. Denote Morse index and zero dimension of 2 Journal of Function Spaces u 0 by μðu 0 Þ and νðu 0 Þ, respectively. The following propositions are essential tools to verify our main result.
Proposition 3 [12]. Let I satisfy the ðDÞ condition. We have (J 1 ) if C q ðI,∞Þ ≇ 0 holds for some q; then, I must have a critical point x such that C q ðI, xÞ ≇ 0; (J 2 ) if 0 is the isolated critical point of I and C q ðI,∞Þ ≇ C q ðI, 0Þ holds for some q; then, I has a nonzero critical point.
Proposition 4 [18]. Suppose u 0 is the isolated critical point of I and I ′ ′ðu 0 Þ is a Fredholm operator. Further, if μðu 0 Þ and νðu 0 Þ are finite, there holds (J 3 ) if u 0 is the local minimum point of I, then Proposition 5 [18]. Let A : S ⟶ S be a self-adjoint linear operator with the isolated spectral point 0 and write I in the form of where Q ∈ C 1 ðS, ℝÞ such that lim Suppose f satisfies the ðDÞ condition and I satisfies the angle condition at infinity: ðAC ± ∞ Þ there exist constants M > 0 and α ∈ ð0, 1Þ such that Then, where In our proofs, we also need the following Mountain Pass Lemma.
Proposition 6 [22]. Let S be a real Banach space and I ∈ C 1 ðS, ℝÞ satisfy the Palais-Smale (PS in short) condition. Further, if Then, I possesses a critical value c ≥ a given by where Denoted by λ k be the k-th eigenvalue corresponding to linear eigenvalue problem of the equation (1), namely, We claim Proof. Rewrite u as where · tr denotes the transpose of vector·. Then, the functional I, defined by (7), can be expressed by where At first, it is easy to get that A possesses at most T 1 T 2 eigenvalues. Subsequently, we need to prove that A is a positive definite matrix.

Main Result and Its Proof
Thanks to above preparations, we are ready to establish our main result and state the detailed proof of it.
First, we give some notations.
and gðði, jÞ, uÞ = f ðði, jÞ, uÞ − λ k u. Then, it yields that We also need the following denotation. ðg ± Þ If ku n k ⟶ ∞ such that kv n k/ku n k ⟶ 1 as n ⟶ ∞, then there exist δ > 0 and N ∈ ℕ such that where u n = v n + w n , v n ∈ W 0 , w n ∈ W + ⊕ W − . Now we state our main result as the following.
According to Proposition 3, we are to verify the compactness conditions (the ðCÞ c condition) of I under the assumptions given in our theorem. Proof. Suppose that there exists a sequence fu m g ⊆ S such that Due to ðS, h·, · iÞ is a T 1 T 2 -dimensional real Hilbert space, it suffices to verify that fu m g is bounded. Arguing indirectly, suppose fu m g is unbounded, namely, Denote u m = u m /ku m k, then k u m k = 1. As a result, there is a convergent subsequence for f u m g. It might as well be set as itself, and there exists u ∈ S, k uk = 1 such that u m ⟶ u. Recall gðði, jÞ, uÞ = f ðði, jÞ, uÞ − λ k u, then for all φ ∈ S, we have Meanwhile, (25) implies that gðði, jÞ, u m Þ/ku m k ⟶ 0 as m ⟶ ∞. Thus, (29) gives Hence, Journal of Function Spaces Therefore, we can deduce According to ðg + Þ, there exist δ > 0 and N ∈ ℕ such that Making use of (30), we get Furthermore, since u m = v m + w m , it follows that which contradicts the hypothesis. Therefore, fu m g is bounded.
Now, we will calculate its critical groups at infinity, C q ðI, ∞Þ, via Proposition 5. Proof. Since S is a T 1 T 2 -dimensional Hilbert space and A is a positive definite matrix, there exists a self-adjoint linear operator, which can be still represented by A. Write Then, which has the form (12) with QðuÞ = −GðuÞ. According to (25), QðuÞ satisfies Moreover, I satisfies the ðCÞ condition guarantees that the ðDÞ condition is fulfilled and ker ðA − JÞ = spanfϕ k g. Subsequently, what we need to do is to show ðAC − ∞ Þ is met with the condition ðg + Þ. Otherwise, for every natural number m and every α m = 1/m, there exists By ðg + Þ, there exist δ > 0 and N ∈ ℕ such that Therefore, Meanwhile, the assumption ðAC − ∞ Þ indicates that which is inconsistent with (42). As a result, I satisfy ðA C − ∞ Þ condition.
In order to gain mountain pass type critical points, we need the following Lemmas.

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satisfies the ðPSÞ condition.

Lemma 13.
Under the condition of Theorem 8, the functional I + possesses a positive critical point u + and C q ðI + , u + Þ ≅ δ q,1 ℤ , the functional I − possesses a negative critical point u − and C q ðI − , u − Þ ≅ δ q,1 ℤ.
Proof. We only prove the case of I + and the proof of the case of I − can be obtained similarily. First of all, we need to prove that I + satisfies the Proposition 6 so that it has a nonzero critical point u + . As a matter of fact, I + ð0Þ = 0 and according 6 Journal of Function Spaces to the conclusion of Lemma 11, I + satisfies the ðPSÞ condition. Now, we must prove that I + satisfies ðJ 4 Þ and ðJ 5 Þ.
Proof of Theorem 14. Let ð _ 1Þ [ð _ 1 _ 1Þ] holds. On one hand, through Lemma 10 we get On the other hand, because of F ′ ′ð0Þ < λ 1 , I ′ ð0Þ = 0 and we can refer that 0 is the local minimum of I. Moreover, 0 is the isolated critical point of I which implies that I ′′ð0Þ is a Fredholm operator with finite Morse index and zero dimension. From Proposition 4, it gives that For the reason that I satisfies the ðDÞ condition from Lemma 9, we can deduce I possesses a critical point u 1 ≠ 0 such that As a result, we conclude that when k ≥ 2½k ≥ 3, u + , u − , and u 1 are the nontrivial critical points of I. Thus, the proof of Theorem 14 is finished.

An Example
As an application of our result, an example is elaborated here.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.