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.
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.
The following abstract theory has been developed by Chang .
The following properties are known:
Lemma 2.5 (see [19, Mountain Pass Theorem]).
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
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 ). Let be a constant such that . For , , we have .
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.
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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