Multiple Positive Solutions in the Sense of Distributions of Singular BVPs on Time Scales and an Application to Emden-Fowler Equations

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,

(1.1)

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

(1.2)

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:

(1.3)

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. , .

• (C_c) 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

(1.5)

satisfies that is -measurable in .

We consider the spaces

(1.6)

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

(1.7)

where is the set of all absolutely continuous functions on . We denote as

(1.8)

The set is endowed with the structure of Hilbert space together with the inner product given for every by

(1.9)

we denote as its induced norm.

Moreover, we consider the sets

(1.10)

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

(1.11)

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.

(C_g)For every , there exists such that

(2.1)

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

(2.2)

-a.e. on .

We fix a sequence of positive numbers strictly decreasing to zero; for every , we define as

(2.3)

Note that satisfies and ; consider the following modified weak problem

(2.4)

Definition 2.2.

is said to be a weak solution to if , on , and equality

(2.5)

holds for all .

is said to be a weak lower solution to if on , and inequality

(2.6)

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

(2.7)

Proposition 2.4.

Suppose that and satisfy and , satisfies , and satisfies .

Then, if for every , is a weak solution to and

(2.8)

(2.9)

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 ,

(2.10)

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

(2.11)

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

(2.12)

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

(2.13)

thus, (2.2) yields to

(2.14)

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

(2.15)

whence it follows, by taking limits, that

(2.16)

which is equivalent because and on to

(2.17)

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

(2.18)

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

(2.19)

For every , consider as a subspace of by defining it for every as on and define the functional for every as

(2.20)

where function is defined for -a.e. and all as

(2.21)

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 ,

(2.22)

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

(2.23)

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

(2.24)

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

(2.25)

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. ,

(2.26)

so that, by(2.20) and(2.22), we have, for every ,

(2.27)

moreover, as we know that on for some , it follows from that there exists such that

(2.28)

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

(3.1)

(H₂) For every , there exist and such that

(3.2)

(H₃) There are such that

(3.3)

for some , where is the smallest positive eigenvalue of problem

(3.4)

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

(3.5)

satisfies that on and on .

Let be so large that , we obtain, by , that

(3.6)

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

(3.7)

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

(3.8)

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

(3.9)

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

(3.10)

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

(3.11)

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

(3.12)

where is the class of paths in joining and , then there is a sequence such that

(3.13)

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. ,

(3.14)

which implies, by (2.20) and Wirtinger's inequality [10, Corollary 3.2], that there exists a constant such that

(3.15)

Thereby, as , there exist constants such that

(3.16)

Let be arbitrary. From the second relation in , we obtain that

(3.17)

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

(3.18)

hence, Lemma 3.4 establishes the existence of a sequence such that

(3.19)

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

(3.20)

we will show that this sequence is bounded in .

From (2.2), we deduce that

(3.21)

For every , from (2.2), (2.20), and(2.22), we have that

(3.22)

where, for -a.e. ,

(3.23)

as a straight-forward consequence of the convexity of and conditions , , , and(3.17), we deduce that there exist constants and such that

(3.24)

Therefore, relations(3.20), (3.21), (3.22), and (3.24) allow to assert that sequence is bounded in and so, as for every ,

(3.25)

We conclude by(3.20), , and that is bounded in and Proposition 2.6 leads to the result.