- Research Article
- Open Access
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.
- Weak Solution
- Sobolev Space
- Mountain Pass
- Homogeneous Dirichlet Boundary Condition
- Multiple Positive Solution
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.
3.1. 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.
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