# Positive Solutions for Multiparameter Semipositone Discrete Boundary Value Problems via Variational Method

- Jianshe Yu
^{1, 2}Email author, - Benshi Zhu
^{1}and - Zhiming Guo
^{2}

**2008**:840458

https://doi.org/10.1155/2008/840458

© Jianshe Yu et al. 2008

**Received: **13 March 2008

**Accepted: **24 August 2008

**Published: **12 October 2008

## Abstract

We study the existence, multiplicity, and nonexistence of positive solutions for multiparameter semipositone discrete boundary value problems by using nonsmooth critical point theory and subsuper solutions method.

## 1. Introduction

Let and be the set of all integers and real numbers, respectively. For , define , , when .

where are parameters, is a positive integer, is the forward difference operator, , is a continuous positive function satisfying , and is continuous and eventually strictly positive with .

We notice that for fixed , whenever is sufficiently small. We call (1.1) a semipositone problem. Semipositone problems are derived from [1], where Castro and Shivaji initially called them nonpositone problems, in contrast with the terminology positone problems, put forward by Keller and Cohen in [2], where the nonlinearity was positive and monotone. Semipositone problems arise in bulking of mechanical systems, design of suspension bridges, chemical reactions, astrophysics, combustion, and management of natural resources; for example, see [3–6].

In general, studying positive solutions for semipositone problems is more difficult than that for positone problems. The difficulty is due to the fact that in the semipositone case, solutions have to live in regions where the nonlinear term is negative as well as positive. However, many methods have been applied to deal with semipositone problems, the usual approaches are quadrature method, fixed point theory, subsuper solutions method, and degree theory. We refer the readers to the survey papers [7, 8] and references therein.

Due to its importance, in recent years, continuous semipositone problems have been widely studied by many authors, see [9–15]. However, we noticed that there were only a few papers on discrete semipositone problems. One can refer to [16–18]. In these papers, semipositone discrete boundary value problems with one parameter were discussed, and subsuper solutions method and fixed point theory were used to study them. To the authors' best knowledge, there are no results established on semipositone discrete boundary value problems with two parameters. Here we want to present a different approach to deal with this topic. In [11], Costa et al. applied the nonsmooth critical point theory developed by Chang [19] to study the existence and multiplicity results of a class of semipositone boundary value problems with one parameter. We think it is also an efficient tool in dealing with the semipositone discrete boundary value problems with two parameters.

We will prove in Section 3 that the sets of positive solutions of (1.1) and (1.3) do coincide. Moreover, any nonzero solution of (1.3) is nonnegative.

Our main results are as follows.

Theorem 1.1.

Then for fixed , there is a such that for , problem (1.3) has a nontrivial nonnegative solution. Hence problem (1.1) has a positive solution.

Remark 1.2.

which shows that is superlinear at infinity.

Remark 1.3.

Hence is subquadratic at infinity.

Theorem 1.4.

Suppose that the conditions of Theorem 1.1 hold. Moreover, is increasing on . Then there is a such that for , problem (1.1) has at least two positive solutions for sufficiently small .

Theorem 1.5.

Suppose that the conditions of Theorem 1.1 hold. Moreover, is nondecreasing on . Then for fixed , problem (1.1) has no positive solution for sufficiently large .

## 2. Preliminaries

The following abstract theory has been developed by Chang [19].

Definition 2.1.

The following properties are known:

Definition 2.2.

The generalized gradient has the following main properties.

- (1)
- (2)
- (3)

Definition 2.3.

is a critical point of the locally Lipschitz functional if .

Definition 2.4.

is said to satisfy Palais-Smale condition (PS) condition for short) if any sequence such that is bounded and has a convergent subsequence.

Lemma 2.5 (see [19, Mountain Pass Theorem]).

Let be a real Hilbert space and let be a locally Lipschitz functional satisfying (PS) condition. Suppose that and that the following hold.

Definition 2.6.

then is called a subsolution of problem (2.8).

Definition 2.7.

then is called a supersolution of problem (2.8).

Lemma 2.8.

Suppose that there exist a subsolution and a supersolution of problem (2.8) such that in . Then there is a solution of problem (2.8) such that in .

Remark 2.9.

If (2.8) is replaced by (1.1), then we have similar definitions and results as Definitions 2.6, 2.7, and Lemma 2.8

## 3. Proof of Main Results

Remark 3.1.

We can show that for , for . For fixed and sufficiently small , . Then .

Remark 3.2.

Lemma 3.3.

If *u* is a solution of (1.3), then
. Moreover, either
in
, or
everywhere.

Proof.

It is not difficult to see that for . In fact, no matter that or , the former inequality holds. Hence .

Therefore . It follows that everywhere.

Lemma 3.4.

If (1.6) and (1.7) hold, then for large , where .

Proof.

Since is superlinear and is sublinear, . Then . Moreover, since is subquadratic and is superlinear, . Therefore, . From the above results, we can conclude that .

Lemma 3.5.

If (1.6) and (1.7) hold, then satisfies (PS) condition.

Proof.

This implies that is bounded. Since is finite dimensional, has a convergent subsequence in .□

Lemma 3.6.

For fixed , there exist and such that if , then for .

Proof.

Lemma 3.7.

Proof.

Hence there is a such that . Let . Then and . The second condition of Mountain Pass theorem is verified.□

Proof of Theorem 1.1.

Clearly, . Lemma 3.5 implies that satisfies (PS) condition. It follows from Lemmas 3.6, 3.7, and 2.5 that has a nontrivial critical point such that . By Lemma 3.3 and Remark 3.2, is a positive solution of (1.1). The proof is complete.□

Proof of Theorem 1.4.

We will apply the subsuper solutions method to prove the multiplicity results.

Notice that and (see [20]). Let be a constant such that . For , , we have .

that is, is a subsolution of (3.26).

which shows that is a supersolution of (3.26). Therefore, by Lemma 2.8, there is a solution of (3.26) such that .

which implies that is a subsolution of (1.1).

Hence is a supersolution of (1.1). Thus, by Remark 2.9, problem (1.1) has a solution such that for and small, which is positive for .

where . On the other hand, by Lemma 3.6, we can take appropriate such that if , then for . Hence by Theorem 1.1, . So and , which shows that and are two different positive solutions of (1.1). The proof is complete.

Proof of Theorem 1.5.

For , we obtain a contradiction. So for a given , (1.1) has no positive solution if is large. The proof is complete.□

Example 3.8.

We give an example to illustrate the result of Theorem 1.1. Let and . Clearly, and satisfy the conditions of Theorem 1.1. Then problem (1.1) has at least a positive solution.

## Declarations

### Acknowledgments

The authors would like to thank the referees for valuable suggestions. This project is supported by National Natural Science Foundation of China (no. 10625104) and Research Fund for the Doctoral Program of Higher Education of China (Grant no. 20061078002).

## Authors’ Affiliations

## References

- Castro A, Shivaji R: Nonnegative solutions for a class of nonpositone problems.
*Proceedings of the Royal Society of Edinburgh. Section A*1988, 108(3-4):291-302. 10.1017/S0308210500014670MATHMathSciNetView ArticleGoogle Scholar - Keller HB, Cohen DS: Some positone problems suggested by nonlinear heat generation.
*Journal of Mathematics and Mechanics*1967, 16: 1361-1376.MATHMathSciNetGoogle Scholar - Afrouzi GA, Rasouli SH:Population models involving the
-Laplacian with indefinite weight and constant yield harvesting.
*Chaos, Solitons & Fractals*2007, 31(2):404-408. 10.1016/j.chaos.2005.09.067MATHMathSciNetView ArticleGoogle Scholar - Myerscough MR, Gray BF, Hogarth WL, Norbury J: An analysis of an ordinary differential equation model for a two-species predator-prey system with harvesting and stocking.
*Journal of Mathematical Biology*1992, 30(4):389-411.MATHMathSciNetView ArticleGoogle Scholar - Selgrade JF: Using stocking or harvesting to reverse period-doubling bifurcations in discrete population models.
*Journal of Difference Equations and Applications*1998, 4(2):163-183. 10.1080/10236199808808135MATHMathSciNetView ArticleGoogle Scholar - Yao Q:Existence of
solutions and/or positive solutions to a semipositone elastic beam equation.
*Nonlinear Analysis: Theory, Methods & Applications*2007, 66(1):138-150. 10.1016/j.na.2005.11.016MATHMathSciNetView ArticleGoogle Scholar - Castro A, Maya C, Shivaji R: Nonlinear eigenvalue problems with semipositone structure. In Proceedings of the Conference on Nonlinear Differential Equations (Coral Gables, Fla, 1999), October 2000, Electronic Journal of Differential Equations Conference. Volume 5. Southwest Texas State University, San Marcos, Tex, USA; 33-49.Google Scholar
- Lions P-L: On the existence of positive solutions of semilinear elliptic equations.
*SIAM Review*1982, 24(4):441-467. 10.1137/1024101MATHMathSciNetView ArticleGoogle Scholar - Agarwal RP, Grace SR, O'Regan D: Semipositone higher-order differential equations.
*Applied Mathematics Letters*2004, 17(2):201-207. 10.1016/S0893-9659(04)90033-XMATHMathSciNetView ArticleGoogle Scholar - Ali J, Shivaji R:On positive solutions for a class of strongly coupled
-Laplacian systems.
*Proceedings of the International Conference in Honor of Jacqueline Fleckinger, June-July 2007, Toulouse, France, Electronic Journal of Differential Equations Conference*16: 29-34.MathSciNetGoogle Scholar - Costa DG, Tehrani H, Yang J: On a variational approach to existence and multiplicity results for semipositone problems.
*Electronic Journal of Differential Equations*2006, 2006(11):1-10.MathSciNetGoogle Scholar - Dancer EN, Zhang Z:Critical point, anti-maximum principle and semipositone -Laplacian problems. Discrete and Continuous Dynamical Systems 2005, (supplement):209-215.Google Scholar
- Hai DD, Shivaji R:An existence result on positive solutions for a class of
-Laplacian systems.
*Nonlinear Analysis: Theory, Methods & Applications*2004, 56(7):1007-1010. 10.1016/j.na.2003.10.024MATHMathSciNetView ArticleGoogle Scholar - Hai DD, Shivaji R: Uniqueness of positive solutions for a class of semipositone elliptic systems.
*Nonlinear Analysis: Theory, Methods & Applications*2007, 66(2):396-402. 10.1016/j.na.2005.11.034MATHMathSciNetView ArticleGoogle Scholar - Yebari N, Zertiti A: Existence of non-negative solutions for nonlinear equations in the semi-positone case. In Proceedings of the Oujda International Conference on Nonlinear Analysis (Oujda 2005), 2006, San Marcos, Tex, USA, Electronic Journal of Differential Equations Conference. Volume 14. Southwest Texas State University; 249-254.Google Scholar
- Agarwal RP, O'Regan D: Nonpositone discrete boundary value problems.
*Nonlinear Analysis: Theory, Methods & Applications*2000, 39(2):207-215. 10.1016/S0362-546X(98)00183-7MATHMathSciNetView ArticleGoogle Scholar - Agarwal RP, Grace SR, O'Regan D: Discrete semipositone higher-order equations.
*Computers & Mathematics with Applications*2003, 45(6–9):1171-1179.MATHMathSciNetView ArticleGoogle Scholar - Jiang D, Zhang L, O'Regan D, Agarwal RP: Existence theory for single and multiple solutions to semipositone discrete Dirichlet boundary value problems with singular dependent nonlinearities.
*Journal of Applied Mathematics and Stochastic Analysis*2003, 16(1):19-31. 10.1155/S1048953303000029MATHMathSciNetView ArticleGoogle Scholar - Chang K-C: Variational methods for non-differentiable functionals and their applications to partial differential equations.
*Journal of Mathematical Analysis and Applications*1981, 80(1):102-129. 10.1016/0022-247X(81)90095-0MATHMathSciNetView ArticleGoogle Scholar - Agarwal RP:
*Difference Equations and Inequalities: Theory, Methods, and Application, Monographs and Textbooks in Pure and Applied Mathematics*.*Volume 228*. 2nd edition. Marcel Dekker, New York, NY, USA; 2000:xvi+971.Google Scholar

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