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.
(C) (i) For every , is -measurable in ;
(ii) For -a.e. , .
(Cc) For every with , there exists such that(1.4)
Moreover, in order to use variational techniques and critical point theory, we will assume that satisfy the following condition.
satisfies that is -measurable in .
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 denote as its induced norm.
where is the set of all functions such that their restriction to every closed subinterval of belong to the Sobolev space .
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.
holds for all .
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 devoted to prove the existence of multiple positive solutions to by using perturbation and variational methods.
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.
2. Approximation to by Weak Problems
In this section, we will deduce sufficient conditions for the existence of solutions in the sense of distributions to , where and satisfy and , satisfies , and satisfies the following condition.
Under these hypotheses, we will be able to approximate solutions in the sense of distributions to problem by a sequence of weak solutions to weak problems.
First of all, we enunciate a useful property of absolutely continuous functions on whose proof we omit because of its simplicity.
-a.e. on .
holds for all .
holds for all such that on .
The concept of weak upper solution to is defined by reversing the previous inequality.
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.
Assume that satisfy and , satisfies , and satisfies .
Then, if for some there exist and as a lower and an upper weak solution, respectively, to such that on , then has a weak solution .
Suppose that and satisfy and , satisfies , and satisfies .
then a subsequence of converges pointwise in to a solution in the sense of distributions to .
that is, is bounded in and hence, there exists a subsequence which converges weakly in and strongly in to some .
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 , .
Let be given by on for all and so that on , , is continuous in every isolated point of the boundary of , and converges pointwise in to .
we know (see ) that on .
which implies that on and so on . Thereby, the continuity of in every dense point of the boundary of and the arbitrariness of guarantee that .
Finally, we will see that(1.11) holds for every test function ; fix one of them.
and the proof is therefore complete.
Propositions 2.3 and 2.4 lead to the following sufficient condition for the existence of at least one solution in the sense of distributions to problem .
Let be such that satisfy and , satisfies and satisfies .
then has a solution in the sense of distributions .
and weak solutions to match up to the critical points of .
Next, we will assume the following condition.
(NI) For -a.e. , is nonincreasing on .
Suppose that is such that satisfy and , satisfies and , and satisfies .
then has a subsequence convergent pointwise in to a nontrivial function such that in and is a solution in the sense of distributions to .
Since is bounded in , it has a subsequence which converges weakly in and strongly in to some .
which implies, from(2.23), that on and so on .
thus, is a solution in the sense of distributions to .
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
In this section, we will derive the existence of solutions in the sense of distributions to where , is a small parameter, and satisfy , as well as the following conditions.
3.1. Existence of One Solution. Uniqueness
Suppose that satisfy , and . Then, there exists a such that for every , problem with has a solution in the sense of distributions .
Let be arbitrary; conditions guarantee that satisfies . We will show that there exists a such that for every , hypotheses in Corollary 2.5 are satisfied.
satisfies that on and on .
whence it follows that is a weak lower solution to .
holds, which implies that is a weak upper solution to .
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 .
If satisfies , , and , then, with has at most one solution in the sense of distributions.
thus, on . The arbitrariness of leads to on and by interchanging and , we conclude that on .
If satisfies , , , and with , then with has a unique 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.
(H4) For -a.e. , is nonincreasing and convex on with given in .
We will use the following variant of the mountain pass, see .
Let be such that , and hold. Then, there exists an such that for every , problem with has two solutions in the sense of distributions such that on and .
Conditions allow to suppose that for -a.e. , is nonnegative, nonincreasing, and convex on because these conditions can be obtained by simply replacing on and with and , respectively.
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.
we will show that this sequence is bounded in .
We conclude by(3.20), , and that is bounded in and Proposition 2.6 leads to the result.
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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