Abstract
In this paper, we consider the multiplicity of solutions for a discrete boundary value problem involving the singular -Laplacian. In order to apply the critical point theory, we extend the domain of the singular operator to the whole real numbers. Instead, we consider an auxiliary problem associated with the original one. We show that, if the nonlinear term oscillates suitably at the origin, there exists a sequence of pairwise distinct nontrivial solutions with the norms tend to zero. By our strong maximum principle, we show that all these solutions are positive under some assumptions. Moreover, the solutions of the auxiliary problem are solutions of the original one if the solutions are appropriately small. Lastly, we give an example to illustrate our main results.
1. Introduction
Let and denote the sets of integers and real numbers, respectively. For , define and when .
In this paper, we consider the following boundary value problem of prescribed mean curvature equations in Minkowski spaces: where is a given positive integer, is a constant in , is the backward difference operator defined by , is the forward difference operator defined by , and for each .
In 2019, Chen et al. in [1] considered problem (1) in the case where and , that is,
By using upper and lower solutions, the Brouwer degree theory, and Szulkin’s critical point theory for convex, lower semicontinuous perturbations of -functions, the authors obtained the intervals of the parameter such that problem (2) has zero, one, or two positive solutions. Earlier in 2008, Bereanu and Mawhin in [2] obtained the existence of at least one or two solutions for the boundary value problems of second-order nonlinear differences with singular -Laplacian by using the Brouwer degree together with fixed point reformulations. For the existence and multiplicity of positive solutions of the associated differential problems to (1), we refer to [3, 4]. And for the boundary value problems of nonsingular differential equations, we refer to [5–9].
Difference equations arise in various research fields. For the existence and multiplicity of solutions of boundary value problems of difference equations, the classical methods are fixed point theory, the method of upper and lower solution techniques, Rabinowitz’s global bifurcation theorem, etc. (see [2, 10, 11]). Since 2003, variational methods have been employed to study difference equations [12], by which various results are obtained. See, for example, periodic solutions and subharmonic solutions [13, 14], homoclinic solutions [15–22], heteroclinic solutions [23], and boundary value problems [24–27]. In recent years, boundary value problems of difference equations involving -Laplacian have aroused extensive attention from scholars; for example, in 2019, Zhou and Ling in [28] considered the following Dirichlet problem of the second-order nonlinear difference equation: where is the mean curvature operator defined by . The authors obtained the existence of infinitely many positive solutions for problem (3). The authors in [29] extended the results of [28] to the following Dirichlet problem:
For the Robin problem of the second-order nonlinear difference equation, where is a special -Laplacian operator [13] defined by with ; we refer to [30].
Up to now, there is less work on the boundary value problems of difference equations involving the singular -Laplcian; the known results are the existence of at least one or two solutions. The aim of this paper is to obtain the existence of infinitely many solutions for problem (1), and problem (1) contains the Dirichlet problem (when ) and Robin problem (when ) as special cases. The difficulty lies that the domain of the singular -Laplcian is a finite open interval , not . The tool is a critical point result in [31]. However, we can not apply the critical point theory to the problem (1) directly, since the domain of the singular -Laplcian in (1) is ; we need to extend the domain of the singular -Laplcian to . Instead, we consider the auxiliary problem (20) associated with problem (1) in Section 2. We will show that solutions of problem (20) are solutions of problem (1) if the solutions are appropriately small. For general background on difference equations, we refer the reader to monographs [32, 33].
This paper is organized as follows. In Section 2, an auxiliary problem associated with problem (1) is established, the variational framework associated with this auxiliary problem is established, and the abstract critical point theorem is recalled. In Section 3, our main results are presented. And we also establish a strong maximum principle and obtain the existence of infinitely many positive solutions for (1) according to the oscillating behavior of at the origin. Finally, in Section 4, an example is given to illustrate our main results.
2. Preliminaries
In this section, we will first introduce a lemma (Theorem 2.5 of [31]).
Let be a reflexive real Banach space, and let be a function satisfying the following structure hypothesis:
for all , where are two functions of class on with coercive, i.e., , and is a real positive parameter.
If , let
Obviously, . When , in the sequel, we agree to read as .
Lemma 1. Assume that the condition holds, and ; then, for each , the following alternative holds: either
(a1) there is a global minimum of which is a local minimum of , or
(a2) there is a sequence of pairwise distinct critical points (local minima) of , with , which weakly converges to the global minimum of .
We will use this lemma to investigate problem (1). Now, we establish the variational framework associated with problem (1). We consider the -dimensional Banach space. endowed with the norm
We consider another norm in , that is,
For each , there exists a , such that thus,
We mention that the equality in (11) holds if we let , where is a nonzero constant. In fact, in this case, and .
We notice that the singular operator in problem (1) only defined for . In order to use Lemma 1, we need to extend the domain of the singular operator to . Take
Then, is continuous in , and the primary function of is given by
We define for each , where for every . When , we read as 0 in (14), since
Put for . Then, and are two functionals of class whose Gâteaux derivatives at the point are given by for all . It is clear that then,
Consequently, the critical points of in are exactly the solutions of the following boundary value problem:
Remark 2. If is a solution of problem (20) with for , then, is a solution of problem (1).
3. Main Results
First, we consider the existence of nontrivial solutions for problem (1). Let
We have the following:
Theorem 3. Assume that there exist two real sequences and , with and , such that Then, for each , problem (1) admits a sequence of nontrivial solutions which converges to zero.
Remark 4. A sequence in is said to converge to zero if as .
Proof. Take and as defined by (14); we will prove Theorem 3 by using Lemma 1. Since , it is easy to see that , and is satisfied. Put
Since , there is no harm in assuming that ; then, . If and
then, for . Noting that for , we see that for and takes the form
Therefore,
which implies that . By (11), we have
According to the definition of , we have
For each , let be defined by for every and . Then,
by using (22). Thus,
Therefore, by (23), we know that
To get our results, we need to show that conclusion (a2) of Lemma 1 holds. Therefore, we want to show that the global minimum of is not a local minimum of . To prove this, we consider two cases: and . In the case where , let be a sequence of positive numbers, with and , such that
Defining a sequence in by for and , we have
In the case where , since , we can choose a small number such that
By the definition of , we can find a sequence of real numbers with such that and
Defining a sequence in by for and , we have
Noticing that , we see that is not a local minimum of by combining the above two cases. Therefore, by Lemma 1 and Remark 2, we know the conclusion of Theorem 3 holds.
Now, let
Then, there exists a sequence of positive numbers with such that
Taking for all , by Theorem 3, we get the following corollary.
Corollary 5. If Then, for each , problem (1) admits a sequence of nontrivial solutions which converges to zero.
To obtain the positive solutions of problem (20), we need the following strong maximum principle.
Theorem 6. Assume such that either for all . Then, either for all or .
Proof. There exists such that
If , then for all and the proof is complete.
If , then . Because and , is increasing in , and , we have
On the other hand, let ; (40) implies
By combining (42) with (43), we get . That is . If , we have . Otherwise, . Replacing by , we get . Continuing this process times, we have for . Similarly, we have for . Thus, and the proof is complete.
Now, we are ready to establish the existence of positive solutions for problem (1); we have
Corollary 7. If for all ,
Then, for each , problem (1) admits a sequence of positive solutions which converges to zero.
Proof. Put
Noticing that , we see that for all . By Corollary 5, we know that problem (1) with replaced by admits a sequence of nontrivial solutions which converges to zero for each . And by Theorem 6, we know that all these solutions are positive.
4. An Example
In this section, we give an example to illustrate our main results.
Example 8. Consider the boundary value problem (1) with for . Then, Since for , we see that is increasing in . Thus, Let ; then (44) holds. By Corollary 7, for each , problem (1) admits a sequence of positive solutions which converges to zero.
5. Conclusions
In this paper, we consider a discrete boundary value problem involving the singular -Laplacian. The problem contains the Dirichlet problem (when ) and Robin problem (when ) as special cases. Since the domain of the singular operator is , we can not apply the critical point theory to this problem directly. Therefore, we extend the domain of the singular operator to the whole real numbers and consider an auxiliary problem associated with the original one. The conditions for the multiplicity of positive solutions of the discrete boundary problem are found, and an illustrative example is given. The method in this paper provides a new way to discuss the boundary value problems containing a singular Laplacian.
Data Availability
No data were used to support the study.
Conflicts of Interest
The author declares that he has no conflicts of interest.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant No. 11571084).