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

Abstract

In the present paper, we consider a second-order nonlinear partial difference equation with Dirichlet boundary conditions. Applying variational method together with the Morse theory, we establish a criterion to obtain at least three nontrivial solutions. An example is also elaborated to demonstrate our main result.

1. Introduction

Let and are natural number set and integer set, respectively. Denote for any , and . and are the forward difference operators defined by and . Given integers , , write , we deal with self-adjoint partial difference equation of the form subject to Dirichlet boundary conditions

Here, is differentiable in and there exists a function such that for each . Further, for all , throughout this paper, we assume that

,

, , and .

As usual, if satisfies (1.1)-(1.2), we say is a solution of (1) and (2). According to , it is easy to see that (1) and (2) admit a trivial solution . 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 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 , 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 [12–14]. 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 [15–18]. 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.

2. 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 , , define an inner product by then the induced norm is

Hence, is a -dimensional Hilbert space.

Consider the functional as the following form:

Note that is differentiable in , which ensures is twice differentiable. What is more, for any , , 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 on .

In the following, we introduce some basic facts.

Definition 1. [21]. The functional satisfies the weaker Cerami condition ( condition for short) at the level if any sequence satisfying , as has a convergent subsequence. satisfies condition if satisfies condition at any .

Definition 2 [16, 22]. Let be an isolated critical group of with and be a neighborhood of , the group is called the q-th critical group of at . Let . If is bounded from below by and satisfies condition for all . Then, the group is called the q-th critical group of at infinity.

In applications of Morse theory, it is necessary to make the functional satisfy the deformation condition , which is introduced by [23]. And [24] proves that once the functional satisfies the condition, it must satisfy the deformation condition . Let be a real Hilbert space and . Denote Morse index and zero dimension of by and , respectively. The following propositions are essential tools to verify our main result.

Proposition 3 [12]. Let satisfy the condition. We have () if holds for some ; then, must have a critical point such that ;
() if 0 is the isolated critical point of and holds for some ; then, has a nonzero critical point.

Proposition 4 [18]. Suppose is the isolated critical point of and is a Fredholm operator. Further, if and are finite, there holds
() if is the local minimum point of , then

Proposition 5 [18]. Let be a self-adjoint linear operator with the isolated spectral point 0 and write in the form of where such that . Define , and and are finite numbers. Suppose satisfies the condition and satisfies the angle condition at infinity:
there exist constants and such that Then, where , , , .

In our proofs, we also need the following Mountain Pass Lemma.

Proposition 6 [22]. Let be a real Banach space and satisfy the Palais-Smale (PS in short) condition. Further, if and
() there exist constants , such that ,
() there is such that .
Then, possesses a critical value given by where

Denoted by be the k-th eigenvalue corresponding to linear eigenvalue problem of the equation (1), namely,

We claim

Lemma 7. For , the eigenvalue of (17)-(2) is positive.

Proof. Rewrite as where denotes the transpose of vector. Then, the functional , defined by (7), can be expressed by where , is the matrix corresponding to the quadratic form

At first, it is easy to get that possesses at most eigenvalues. Subsequently, we need to prove that is a positive definite matrix.

In fact, for all and , we have

If , according to , there hold

which lead to that for all and there exists some such that . Together with Dirichlet boundary conditions (2), we can deduce that , which show a contradiction with respect to . Therefore, is positive define, which ensures that for all . Without loss of generality, we can rearrange all eigenvalues of as .

3. 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.

Let and , represent an eigenvector corresponding to the eigenvalue . Write , , and ; then, can be split into

Denote

and . Then, it yields that

We also need the following denotation.

If such that as , then there exist and such that where , , .

Now we state our main result as the following.

Theorem 8. Let , (p), and (24) hold. If, for all ,
() and , ,
()
are satisfied. Then, (1) and (2) possess at least three nontrivial solutions, including a positive solution and a negative solution under either with or with is fulfilled.

According to Proposition 3, we are to verify the compactness conditions (the condition) of under the assumptions given in our theorem.

Lemma 9. Let and (24) hold. If holds, then satisfies the (C) condition.

Proof. Suppose that there exists a sequence such that Due to is a -dimensional real Hilbert space, it suffices to verify that is bounded. Arguing indirectly, suppose is unbounded, namely, Denote , then . As a result, there is a convergent subsequence for . It might as well be set as itself, and there exists , such that .
Recall , then for all , we have Meanwhile, (25) implies that as . Thus, (29) gives Denote , where , , and , then according to we get Hence, Therefore, we can deduce According to , there exist and such that Making use of (30), we get Furthermore, since , it follows that which contradicts the hypothesis. Therefore, is bounded.

Now, we will calculate its critical groups at infinity, , via Proposition 5.

Lemma 10. Let and (24) hold, then
() if holds;
() if holds.

Proof. Since is a -dimensional Hilbert space and is a positive definite matrix, there exists a self-adjoint linear operator, which can be still represented by . Write

Then, which has the form (12) with . According to (25), satisfies

Moreover, satisfies the condition guarantees that the condition is fulfilled and . Subsequently, what we need to do is to show is met with the condition . Otherwise, for every natural number and every , there exists , where , such that , . Therefore, it yields that

By , there exist and such that

Therefore,

Meanwhile, the assumption indicates that

which is inconsistent with (42). As a result, satisfy condition.

In order to gain mountain pass type critical points, we need the following Lemmas.

Lemma 11. Let where . If then the functional satisfies the condition.

Proof. Suppose that there exists a sequence satisfying

Since is a -dimensional real Hilbert space, we only need to verify that is bounded. On the contrary, suppose is unbounded, that is, . Denote that , then and exists , such that . For any , we get

Since for every , we get where , . Therefore, which implies that satisfies

Next, we prove . Write , it is necessary to show . If , (51) and (52) imply that

Together with (), it follows that

Thus, which leads to for all . Obviously, it contradicts with , then . Therefore, is bounded.

Similar to Lemma 11, we have

Lemma 12. Let where . If then the functional satisfies the condition.

Lemma 13. Under the condition of Theorem 8, the functional possesses a positive critical point and , the functional possesses a negative critical point and .

Proof. We only prove the case of and the proof of the case of can be obtained similarily. First of all, we need to prove that satisfies the Proposition 6 so that it has a nonzero critical point . As a matter of fact, and according to the conclusion of Lemma 11, satisfies the condition. Now, we must prove that satisfies and .

On one hand, since is satisfied, there exist and satisfying such that

Then,

Therefore, is satisfied. On the other hand, (24) means that there exists , such that

Choose , where is the eigenvector corresponding to , we have

Therefore, there exists a sufficiently large such that for , which makes is fulfilled. By Proposition 6, possesses a critical point .

In the following, we devote to verifying . Due to , there exists a sequence such that as . Then, for all , there holds

Let and , we have

Denote . Now, we need to prove to get . Arguing indirectly, let , then

Applying (), we get which means . Then, contradicts with . Consequently, .

Now, we calculate . On account of for any , we have and there exists such that , which indicates that is a solution of

Therefore, the corresponding Dirichlet eigenvalue problem admits an eigenvalue . According to , 1 is a single eigenvalue, which means that . Combining with is a surjection, is a Fredholm operator, we conclude that

With the aid of preceding preparations, now we are in the position to complete the verification of Theorem 8 via Proposition 3.

Proof of Theorem 14. Let [] holds. On one hand, through Lemma 10 we get On the other hand, because of , and we can refer that 0 is the local minimum of . Moreover, 0 is the isolated critical point of which implies that is a Fredholm operator with finite Morse index and zero dimension. From Proposition 4, it gives that For the reason that satisfies the condition from Lemma 9, we can deduce possesses a critical point such that As a result, we conclude that when , , , and are the nontrivial critical points of . Thus, the proof of Theorem 14 is finished.

4. An Example

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

Example 15. Take , , then for , , , . Consider with the boundary value conditions (2).

According to the expression of , it follows that

It is not difficult to verify that , , , and

Since , , and for each , then the matrix is positive define, and the eigenvalues are given by

Next, we check . If such that with , then there exist and such that

In fact, we can choose , . Since , then

Therefore, all conditions of Theorem 8 are satisfied and (76)-(2) admits at least three nontrivial solutions, which include a positive solution and a negative solution.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 11971126)