Multiple Positive Solutions in the Sense of Distributions of Singular BVPs on Time Scales and an Application to Emden-Fowler Equations
© Ravi P. Agarwal et al. 2008
Received: 21 April 2008
Accepted: 17 August 2008
Published: 27 August 2008
This paper is devoted to using perturbation and variational techniques to derive some sufficient conditions for the existence of multiple positive solutions in the sense of distributions to a singular second-order dynamic equation with homogeneous Dirichlet boundary conditions, which includes those problems related to the negative exponent Emden-Fowler equation.
Recently, existence theory for positive solutions of second-order boundary value problems on time scales has received much attention (see, e.g., [3–6] for general case,  for the continuous case, and  for the discrete case).
where we say that a property holds for -a.e. or a.e. on -a.e., whenever there exists a set with null Lebesgue -measure such that this property holds for every , is an arbitrary time scale, subindex means intersection to , are such that , , , , , , and is an -Carathéodory function on compact subintervals of , that is, it satisfies the following conditions.
where is the set of all continuous functions on such that they are -differentiable on and their -derivatives are rd-continuous on , is the set of all continuous functions on that vanish on the boundary of , and is the set of all continuous functions on with compact support on . We denote as the norm in , that is, the supremum norm.
We refer the reader to [9–11] for an introduction to several properties of Sobolev spaces and absolutely continuous functions on closed subintervals of an arbitrary time scale, and to  for a broad introduction to dynamic equations on time scales.
From the density properties of the first-order Sobolev spaces proved in [9, Seccion 3.2], we deduce that if is solution in the sense of distributions, then, (1.11) holds for all .
This paper is organized as follows. In Section 2, we deduce sufficient conditions for the existence of solutions in the sense of distributions to . Under certain hypotheses, we approximate solutions in the sense of distributions to problem by a sequence of weak solutions to weak problems. In Section 3, we derive some sufficient conditions for the existence of at least one or two positive solutions to .
These results generalize those given in  for , where problem is defined on the whole interval and the authors assume that instead of and . The sufficient conditions for the existence of multiple positive solutions obtained in this paper are applied to a great class of bounded time scales such as finite union of disjoint closed intervals, some convergent sequences and their limit points, or Cantor sets among others.
We remark that the density properties of the first-order Sobolev spaces proved in [9, Seccion 3.2] allows to assert that relations in Definition 2.2 are valid for all and for all such that on , respectively.
By standard arguments, we can prove the following result.
For every , by considering for each the weak solution to and by repeating the previous construction, we obtain a sequence which converges weakly in and strongly in to some with . By definition, we know that for all , .
we know (see ) that on .
and the proof is therefore complete.
Next, we will assume the following condition.
and hence, since is bounded in and converges pointwise in to the trivial function , we deduce, from the second relation in(2.23) and(2.24), that which contradicts the first relation in(2.23). Therefore, is a nontrivial function.
3. Results on the Existence and Uniqueness of Solutions
3.1. Existence of One Solution. Uniqueness
Therefore, for every so large, we have a lower and an upper solution to , respectively, such that (2.2) is satisfied and so, Corollary 2.5 guarantees that problem has at least one solution in the sense of distributions .
3.2. Existence of Two Ordered Solutions
Next, by using Theorem 3.1 which ensures the existence of a solution in the sense of distributions to , we will deduce, by applying Proposition 2.6, the existence of a second one greater than or equal to the first one on the whole interval ; in order to do this, we will assume that satisfy , , as well as the following conditions.
We will use the following variant of the mountain pass, see .
Let be a solution in the sense of distributions to , its existence is guaranteed by Theorem 3.1, and let be arbitrary; it is clear that with satisfies hypothesis in Proposition 2.6; we will derive the existence of an such that for every , we are able to construct a sequence in the conditions of Proposition 2.6.
This research is partially supported by MEC and F.E.D.E.R. Project MTM2007-61724, and by Xunta of Galicia and F.E.D.E.R. Project PGIDIT05PXIC20702PN, Spain.
- Wong JSW: On the generalized Emden-Fowler equation. SIAM Review 1975, 17(2):339-360. 10.1137/1017036MATHMathSciNetView ArticleGoogle Scholar
- Agarwal RP, O'Regan D, Lakshmikantham V, Leela S: An upper and lower solution theory for singular Emden-Fowler equations. Nonlinear Analysis: Real World Applications 2002, 3(2):275-291. 10.1016/S1468-1218(01)00029-3MATHMathSciNetView ArticleGoogle Scholar
- Agarwal RP, Otero-Espinar V, Perera K, Vivero DR: Existence of multiple positive solutions for second order nonlinear dynamic BVPs by variational methods. Journal of Mathematical Analysis and Applications 2007, 331(2):1263-1274. 10.1016/j.jmaa.2006.09.051MATHMathSciNetView ArticleGoogle Scholar
- Agarwal RP, Otero-Espinar V, Perera K, Vivero DR: Multiple positive solutions of singular Dirichlet problems on time scales via variational methods. Nonlinear Analysis: Theory, Methods & Applications 2007, 67(2):368-381. 10.1016/j.na.2006.05.014MATHMathSciNetView ArticleGoogle Scholar
- Du Z, Ge W: Existence of multiple positive solutions for a second-order Sturm-Liouville-like boundary value problem on a measure chain. Acta Mathematicae Applicatae Sinica 2006, 29(1):124-130.MathSciNetGoogle Scholar
- Khan RA, Nieto JJ, Otero-Espinar V: Existence and approximation of solution of three-point boundary value problems on time scale. Journal of Difference Equations and Applications 2008, 14(7):723-736. 10.1080/10236190701840906MATHMathSciNetView ArticleGoogle Scholar
- Agarwal RP, Perera K, O'Regan D: Positive solutions in the sense of distributions of singular boundary value problems. Proceedings of the American Mathematical Society 2008, 136(1):279-286. 10.1090/S0002-9939-07-09105-8MATHMathSciNetView ArticleGoogle Scholar
- Tian Y, Du Z, Ge W: Existence results for discrete Sturm-Liouville problem via variational methods. Journal of Difference Equations and Applications 2007, 13(6):467-478. 10.1080/10236190601086451MATHMathSciNetView ArticleGoogle Scholar
- Agarwal RP, Otero-Espinar V, Perera K, Vivero DR: Basic properties of Sobolev's spaces on time scales. Advances in Difference Equations 2006, 2006:-14.Google Scholar
- Agarwal RP, Otero-Espinar V, Perera K, Vivero DR: Wirtinger's inequalities on time scales. Canadian Mathematical Bulletin 2008, 51(2):161-171. 10.4153/CMB-2008-018-6MATHMathSciNetView ArticleGoogle Scholar
- Cabada A, Vivero DR: Criterions for absolute continuity on time scales. Journal of Difference Equations and Applications 2005, 11(11):1013-1028. 10.1080/10236190500272830MATHMathSciNetView ArticleGoogle Scholar
- Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Application. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleGoogle Scholar
- Cerami G: An existence criterion for the critical points on unbounded manifolds. Istituto Lombardo. Accademia di Scienze e Lettere. Rendiconti. A 1978, 112(2):332-336.MATHMathSciNetGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.