Positive Solutions for Multiparameter Semipositone Discrete Boundary Value Problems via Variational Method
© Jianshe Yu et al. 2008
Received: 13 March 2008
Accepted: 24 August 2008
Published: 12 October 2008
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.
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 , where Castro and Shivaji initially called them nonpositone problems, in contrast with the terminology positone problems, put forward by Keller and Cohen in , 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 , Costa et al. applied the nonsmooth critical point theory developed by Chang  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.
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.
which shows that is superlinear at infinity.
Hence is subquadratic at infinity.
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 .
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 .
The following abstract theory has been developed by Chang .
The following properties are known:
is subadditive, positively homogeneous, continuous, and convex;
The generalized gradient has the following main properties.
For all , is a nonempty convex and -compact subset of ;
for all .
- (3)If are locally Lipschitz functional, then(2.4)
For any ,
If is a convex functional, then coincides with the usual subdifferential of in the sense of convex analysis.
If is Gâteaux differential at every point of of a neighborhood of and the Gâteaux derivative is continuous, then
- (7)The function(2.5)
exists, that is, there is a such that .
If has a minimum at , then .
is a critical point of the locally Lipschitz functional if .
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.
There exist constants and such that if .
There is an such that and .
then is called a subsolution of problem (2.8).
then is called a supersolution of problem (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 .
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
We can show that for , for . For fixed and sufficiently small , . Then .
If u is a solution of (1.3), then . Moreover, either in , or everywhere.
It is not difficult to see that for . In fact, no matter that or , the former inequality holds. Hence .
Therefore . It follows that everywhere.
If (1.6) and (1.7) hold, then for large , where .
Since is superlinear and is sublinear, . Then . Moreover, since is subquadratic and is superlinear, . Therefore, . From the above results, we can conclude that .
If (1.6) and (1.7) hold, then satisfies (PS) condition.
This implies that is bounded. Since is finite dimensional, has a convergent subsequence in .□
For fixed , there exist and such that if , then for .
There is an such that and .
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.
with and .
Notice that and (see ). 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.□
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.
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).
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