Abstract
In this paper, we consider the existence and multiplicity of solutions for a discrete Dirichlet boundary value problem involving the -Laplacian. By using the critical point theory, we obtain the existence of infinitely many solutions under some suitable assumptions on the nonlinear term. Also, by our strong maximum principle, we can obtain the existence of infinitely many positive solutions.
1. Introduction
Let be a positive integer and denote with the discrete set . In this paper, we consider the existence of infinitely many solutions for the following discrete Dirichlet boundary value problem
where is the discrete -Laplacian, with is the forward difference operator, is continuous for each , , is a positive parameter, and for all
In the past decades, there has been tremendous interest in the study of difference equations, with the development of engineering, physics, economy, and so on (see [1–4]). Most results about the boundary value problems of difference equations are obtained by using the method of upper and lower solutions and fixed point methods (see [5–7]). In 2003, Guo and Yu [8] first applied the critical point theory to study the existence of periodic and subharmonic solutions for a second-order difference equation. Since then, the critical point theory has been employed to study difference equations, and many meaningful results have been obtained, concerning periodic solutions [9, 10], homoclinic solutions [11–13], heteroclinic solutions [14], and especially in boundary value problems [15–20]. For example, Candito and Giovannelli [21] established the existence of multiple solutions of the following problem
Later, Bonanno and Candito [22] established the existence of infinitely many solutions of the following problem
where for all Obviously, (2) is a special case () of (3). After that, under different conditions, D’Aguì et al. [23] established the existence of at least two positive solutions of (3).
In [24], Li and Zhou considered the following discrete mixed boundary value problem
where for all By using the critical point theory, the authors obtained the existence of at least two positive solutions for (4).
The boundary value problems involving the sum of a -Laplacian operator and of a -Laplacian operator is more common, because this arises in the study of stationary solutions of reaction-diffusion systems (see [25]). For example, Mugnai and Papageorgiou [26] and Marano et al. [27] investigated the following Dirichlet problem
where satisfies Carathéodory’s conditions, and they obtained the existence of multiple solutions of (5).
In [28], Nastasi et al. proved the existence of at least two positive solutions for problem (1). Compared with the discrete boundary value problem involving -Laplacian operator, there are few results on the discrete boundary value problem with -Laplacian operator except [28]. Inspired by the above results, we want to investigate the multiplicity of solutions for problem (1).
In this paper, under suitable assumptions, we use the critical point theory obtained in [29] to establish the existence of infinitely many solutions for discrete -Laplacian equations with Dirichlet type boundary conditions. Moreover, by our strong maximum principle, we can obtain the existence of infinitely many positive solutions of (1).
The rest of this paper is organized as follows. In Section 2, we recall the critical point theory and show some basic lemmas. In Section 3, our main results and proofs are presented. After that, we have two examples to explain our main results. We conclude our results in the last section.
2. Preliminaries
Let be a reflexive real Banach space and let be a function satisfying the following structure hypothesis:
(H) for all , where are two functions of class on with coercive, i.e., , and is a real positive parameter
Provided that , put and
There is no doubt that and . When (or ), in the sequel, we agree to regard (or ) as .
Now, we recall Theorem 2.1 of [29], which is our main tool for investigating problem (1).
Lemma 1. Assume that the condition (H) holds. We have
For every and every , the restriction of the functional to admits a global minimum, which is a critical point (local minimum) of in .
If then, for each , the following alternative holds: either
possesses a global minimum, or
There is a sequence of critical points (local minimum) of such that
If then, for each , the following alternative holds: either
There is a global minimum of which is a local minimum of , or
There is a sequence of pairwise distinct critical points (local minima) of , with , which weakly converges to a global minimum of
Here, we consider the -dimensional Banach space and define the norm where , with for all , and . Then, let be endowed with the norm We denote the usual sup-norm by , and then we consider the inequality (see ([30], Lemma 2.2)):
Lemma 2. Let . The following inequalities hold
Proof. The left-hand side of (11) follows by [30]. Consider the right-hand inequality,
Put where the function is given by .
Clearly, and we have the following Gâteaux derivatives at the point : for all Now, for ,
If we plug this result back into the calculation of Gâteaux derivatives above, then for all Let
Consider the functional given as
We have for all Thus, is a solution of problem (1) if and only if is a critical point of .
Lemma 3. Fix such that either for all . Then, either in or .
Proof. Fix and If , then, . Now, if , we can get which implies that Thus, Since and are both strictly increasing, we have , which implies . It follows that , then An easy induction gives That is , and this is absurd. Next, we assume that , Due to the monotonicity of and , , which means . Because , we have . By repeating this argument, it is easy to see which leads to a contradiction.
Now, consider the function given as where . Now, we define , where Similarly, the critical points of are the solutions of the following problem
Lemma 4. If for all , then each nonzero critical point of is a positive solution of (1).
Proof. We note that each positive solution of (31) is a positive solution of (1). By an application of Lemma 3, we conclude that . It follows that the nonzero solutions of (31) are positive and hence are positive solutions of (1).
3. Main Results
Let
The main results are as follows.
Theorem 5. Assume that , and there are two real sequences and , with , such that Then for each , problem (1) admits an unbounded sequence of solutions.
Proof. Fix, then, we can take the real Banach space as defined in Section 2, and the definitions of are the same as before. We will prove Theorem 5 by applying Lemma 1 part (b) to function . Since (H) is trivial to prove, it suffices to prove and turns out to be unbounded from below. To this end, let
Since, owing to (10), if then , and if then . So, let . From , we have .
We obtain
Then, we define such that for every , Clearly and owing to (33). One has
Therefore, . It remains to show that is unbounded from below.
Let be a sequence with for such that . Because , fix such that , and we deduce that there is such that for all . Moreover, since is a continuous function, there exists a constant such that for all . Thus, for all and . It follows that
where . Since , one has
As , it is obvious that . Hence, is unbounded from below and the proof is complete.
Let
The following theorem can be obtained if we change some of the conditions.
Theorem 6. Assume that there are two real sequences and , with , such that (33) holds and Then, for each , problem (1) admits an unbounded sequence of solutions.
Proof. The first half of the argument is analogous to that in Theorem 5, and put as above. So, we have .
Our task now is to verify that is unbounded from below. First, we assume that Fix such that , and let be a sequence with and such that
Taking the sequence in defined by for every , , we have
It is easy to see .
Then, we assume that and fix such that . Let be a sequence with , such that and
Let the sequence in be the same as the case where , such that
which implies that .
So, in both cases, is unbounded from below, which completes the proof of Theorem 6.
Let
Applying part (c) of Lemma 1, we get the following theorem.
Theorem 7. Assume that there exist two real sequences , with , such that Then, for each , problem (1) admits a sequence of nonzero solutions which converges to zero.
Proof. Fix in , and we can take the real Banach space and functional as defined in Section 2. Our aim is to apply Lemma 1 part (c) to function . To this end, let
Owing to (10), if then , and if then . So, let . It follows that if , then . We obtain
Now, for each , let be defined by for every , Clearly , and from (47). We have
Hence, follows.
In fact, , so our task now is to verify that the is not a local minimum of . First, assume that Fix such that , and let be a sequence of positive numbers, with and such that
Thus, taking the sequence in , let for every , . Some tedious manipulation yields
which implies that .
Then, we assume that and fix such that . Let be a sequence of positive numbers, with , such that and
Choosing the same in as the case , one has
That is . Since 0 is the global minimum of , in both cases, is not a local minimum of and the proof is complete.
By setting we get the following consequences.
Corollary 8. Assume that Then, for each , problem (1) admits an unbounded sequence of solutions.
Proof. Let be a sequence of positive numbers with , such that After simple scaling and calculation, we have Taking for each , from Theorem 6, the conclusion follows.
If satisfies the nonnegative condition, we have the following conclusion.
Corollary 9. Assume that for all , and Then, for each , problem (1) admits an unbounded sequence of positive solutions.
Proof. Let
Since ,
for all . From Corollary 8, we know that problem (1) with replaced by admits an unbounded sequence of solutions for each . Then, all these solutions are positive solutions of problem (1) by Lemma 4.
Let
Arguing as in the proof of Corollary 8 and taking for each , by Theorem 7, we have the following corollary.
Corollary 10. Assume that Then, for each , problem (1) admits a sequence of nonzero solutions which converges to zero.
Arguing as in Corollary 9, we have the following result.
Corollary 11. Assume that for all , and Then, for each , problem (1) admits a sequence of positive solutions which converges to zero.
Finally, we give two easy examples to illustrate our results.
Example 1. Let for each . Then, and By choosing , we have From the above calculation, we obtain It is clear that , by Corollary 9, the problem admits an unbounded sequence of positive solutions.
Example 2. Let and for each . Then, for . Since for , is increasing. We have Let be a sufficiently small constant, such that Then, by Corollary 11, for each , problem (1) admits a sequence of positive solutions which converges to zero.
4. Conclusions
In this paper, we consider a discrete Dirichlet boundary value problem involving the -Laplacian. Unlike the existing result in [28], which is the existence of at least two positive solutions, we consider the existence of infinitely many solutions for problem (1) for the first time. In fact, by using Theorem 2.1 of [29], we show that problem (1) admits a sequence of pairwise distinct solutions under some appropriate assumptions on the nonlinear term near at infinity and at the origin. Moreover, we prove the existence of infinitely many positive solutions through our strong maximum principle. It seems that we can use the method in this paper to study other similar problems, such as the existence and multiplicity of solutions for difference equations with different boundary value conditions. This will be left as our future work.
Data Availability
No data were used to support the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant nos. 11971126, 11771104) and the Program for Changjiang Scholars and Innovative Research Team in University (Grant no. IRT 16R16).