Advances in Difference Equations volume 2008, Article number: 796851 (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.
The Emden-Fowler equation,
arises in the study of gas dynamics and fluids mechanics, and in the study of relativistic mechanics, nuclear physics, and chemically reacting system (see, e.g., [1] and the references therein) for the continuous model. The negative exponent Emden-Fowler equation (
) has been used in modelling non-Newtonian fluids such as coal slurries [2]. The physical interest lies in the existence of positive solutions. We are interested in a broad class of singular problem that includes those related with (1.1) and the more general 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, [7] for the continuous case, and [8] for the discrete case).
In this paper, we consider the second-order dynamic equation with homogeneous Dirichlet boundary conditions:
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
Moreover, in order to use variational techniques and critical point theory, we will assume that 
satisfy the following condition.
(PM) For every 
, function 
defined for 
-a.e. 
and all 
, as
satisfies that 
is 
-measurable in 
.
We consider the spaces
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.
On the other hand, we consider the first-order Sobolev spaces
where 
is the set of all absolutely continuous functions on 
. We denote as
The set 
is endowed with the structure of Hilbert space together with the inner product 
given for every 
by
we denote as 
its induced norm.
Moreover, we consider the sets
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 [12] for a broad introduction to dynamic equations on time scales.
Definition 1.1.
is said to be a solution in the sense of distributions to 
if 
, 
on 
, and equality
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 [7] 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.
by Weak ProblemsIn 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.
(Cg)For every 
, there exists 
such that
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.
Lemma 2.1.
If 
, then 
-a.e. on 
.
We fix 
a sequence of positive numbers strictly decreasing to zero; for every 
, we define 
as
Note that 
satisfies 
and 
; consider the following modified weak problem
Definition 2.2.
is said to be a weak solution to 
if 
, 
on 
, and equality
holds for all 
.
is said to be a weak lower solution to 
if 
on 
, and inequality
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.
Proposition 2.3.
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 
.
Next, we will deduce the existence of one solution in the sense of distributions to 
from the existence of a sequence of weak solutions to 
. In order to do this, we fix 
two sequences such that 
is strictly decreasing to 
if 
, 
for all 
if 
and 
is strictly increasing to 
if 
, 
for all 
if 
. We denote that 
, 
. Moreover, we fix 
a sequence of positive numbers strictly decreasing to zero such that
Proposition 2.4.
Suppose that 
and 
satisfy 
and 
, 
satisfies 
, and 
satisfies 
.
Then, if for every 
, 
is a weak solution to 
and
then a subsequence of 
converges pointwise in 
to a solution in the sense of distributions 
to 
.
Proof.
Let 
be arbitrary; we deduce, from (2.2), (2.7), (2.8), and (2.9), that there exists a constant 
such that for all 
,
Therefore, for all 
so large that 
, as 
is a weak solution to 
, by taking 
as the test function in(2.5), from (2.9), 
and 
, we can assert that there exists 
such that
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 will show that 
; we only have to prove that 
is continuous in every dense point of the boundary of 
. Let 
be arbitrary, it follows from 
and 
that there exist 
such that 
on 
and 
for 
-a.e. 
and all 
; let 
be the weak solution to
we know (see [4]) that 
on 
.
For all 
so large that 
, since 
and 
are weak solutions to some problems, by taking 
as the test function in their respective problems, we obtain
thus, (2.2) yields to
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.
For all 
so large that 
and all 
so large that 
, as 
is a weak solution to 
, by taking 
as the test function in (2.5) and bearing in mind(2.7), we have
whence it follows, by taking limits, that
which is equivalent because 
and 
on 
to
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 
.
Corollary 2.5.
Let 
be such that 
satisfy 
and 
, 
satisfies 
and 
satisfies 
.
Then, if for each 
there exist 
and 
a lower and an upper weak solution, respectively, to 
such that 
on 
and
then 
has a solution in the sense of distributions 
.
Finally, fixed 
is a solution in the sense of distributions to 
with 
, we will derive the existence of a second solution in the sense of distributions to 
greater than or equal to 
on 
. For every 
, consider the weak problem
For every 
, consider 
as a subspace of 
by defining it for every 
as 
on 
and define the functional 
for every 
as
where function 
is defined for 
-a.e. 
and all 
as
As a consequence of Lemma 2.1, we deduce that every weak solution to 
is nonnegative on 
and by reasoning as in [4, Section 3], one can prove that 
is weakly lower semicontinuous, 
is continuously differentiable in 
, for every 
,
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 
.
Proposition 2.6.
Suppose that 
is such that 
satisfy 
and 
, 
satisfies 
and 
, and 
satisfies 
.
If 
, 
is a bounded sequence in 
such that
then 
has a subsequence convergent pointwise in 
to a nontrivial function 
such that 
in 
and 
is a solution in the sense of distributions to 
.
Proof.
Since 
is bounded in 
, it has a subsequence which converges weakly in 
and strongly in 
to some 
.
For every 
, by (2.2), we obtain
which implies, from(2.23), that 
on 
and so 
on 
.
In order to show that 
is a solution in the sense of distributions to 
, fix 
arbitrary and choose 
so large that 
, bearing in mind that 
is a solution in the sense of distributions to 
, and the pass to the limit in(2.22) with 
and 
yields to
thus, 
is a solution in the sense of distributions to 
.
Finally, we will see that 
is not the trivial function; suppose that 
on 
. Condition 
ensures that function 
defined in (2.21) satisfies for every 
and 
-a.e. 
,
so that, by(2.20) and(2.22), we have, for every 
,
moreover, as we know that 
on 
for some 
, it follows from 
that there exists 
such that
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.
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.
(H1) There exists a constant 
and a nontrivial function 
such that 
-a.e. on 
and
(H2) For every 
, there exist 
and 
such that
(H3) There are 
such that
for some 
, where 
is the smallest positive eigenvalue of problem
Theorem 3.1.
Suppose that 
satisfy 
, 
and 
. Then, there exists a 
such that for every 
, problem 
with 
has a solution in the sense of distributions 
.
Proof.
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.
Let 
and 
be given in 
, we know, from [4, Proposition 2.7], that we can choose 
so small that the weak solution 
to
satisfies that 
on 
and 
on 
.
Let 
be so large that 
, we obtain, by 
, that
whence it follows that 
is a weak lower solution to 
.
As a consequence of 
, 
and 
, by reasoning as in [4, Theorem 4.2], we deduce that problem
has some weak solution 
which, from Lemma 2.1 and 
, satisfies that 
on 
. We will see that 
is bounded in 
, by taking 
as the test function, we know from (2.2), 
and 
that there exist 
such that
so that, it follows from the fact that the immersion from 
into 
is compact, see [9, Proposition 3.7], Wirtinger's inequality [10, Corollary 3.2] and relation 
that 
is bounded in 
and, hence, 
is bounded in 
. Thereby, condition 
allows to assert that there exists 
, such that for all 
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 
.
Theorem 3.2.
If 
satisfies 
, 
, and 
, then, 
with 
has at most one solution in the sense of distributions.
Proof.
Suppose that 
has two solutions in the sense of distributions 
. Let 
be arbitrary, take 
as the test function in (1.11), by (2.2) and 
, we have
thus, 
on 
. The arbitrariness of 
leads to 
on 
and by interchanging 
and 
, we conclude that 
on 
.
Corollary 3.3.
If 
satisfies 
, 
, 
, and 
with 
, then 
with 
has a unique solution in the sense of distributions.
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 
.
(H5) There are constants 
, 
and 
such that
We will use the following variant of the mountain pass, see [13].
Lemma 3.4.
If 
is a continuously differentiable functional defined on a Banach space 
and there exist 
such that
where 
is the class of paths in 
joining 
and 
, then there is a sequence 
such that
Theorem 3.5.
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 
.
Proof.
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.
For every 
and 
, as a straight-forward consequence of 
, 
, 
, and the compact immersion from 
into 
, we deduce that there exist two constants 
such that function 
, defined in (2.21), satisfies for 
-a.e. 
,
which implies, by (2.20) and Wirtinger's inequality [10, Corollary 3.2], that there exists a constant 
such that
Thereby, as 
, there exist constants 
such that
Let 
be arbitrary. From the second relation in 
, we obtain that
for some constant 
; thus, it is not difficult to prove that there is a 
such that 
on 
, 
and 
and hence, since 
, by denoting as 
the class of paths in 
joining 
and 
, it follows from (3.16) that
hence, Lemma 3.4 establishes the existence of a sequence 
such that
Consequently, bearing in mind that 
and 
for all 
and by removing a finite number of terms if it is necessary, we obtain a sequence 
such that 
for every 
and
we will show that this sequence is bounded in 
.
From (2.2), we deduce that
For every 
, from (2.2), (2.20), and(2.22), we have that
where, for 
-a.e. 
,
as a straight-forward consequence of the convexity of 
and conditions 
, 
, 
, and(3.17), we deduce that there exist constants 
and 
such that
Therefore, relations(3.20), (3.21), (3.22), and (3.24) allow to assert that sequence 
is bounded in 
and so, as for every 
,
We conclude by(3.20), 
, and 
that 
is bounded in 
and Proposition 2.6 leads to the result.
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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.
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Agarwal, R.P., Otero-Espinar, V., Perera, K. et al. Multiple Positive Solutions in the Sense of Distributions of Singular BVPs on Time Scales and an Application to Emden-Fowler Equations. Adv Differ Equ 2008, 796851 (2008). https://doi.org/10.1155/2008/796851
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DOI: https://doi.org/10.1155/2008/796851