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Multiple Positive Solutions in the Sense of Distributions of Singular BVPs on Time Scales and an Application to Emden-Fowler Equations
Advances in Difference Equations volume 2008, Article number: 796851 (2008)
Abstract
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.
1. Introduction
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
(1.4)
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.
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.
(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.
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.
(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

3.1. Existence of One Solution. Uniqueness
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.
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
.
(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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Acknowledgments
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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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
Keywords
- Weak Solution
- Sobolev Space
- Mountain Pass
- Homogeneous Dirichlet Boundary Condition
- Multiple Positive Solution