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fference EquationsVolume 2008, Article ID 796851, 13 pages doi:10.1155/2008/796851 Research Article Multiple Positive Solutions in the Sense of Distributions of Singular BVPs on Time Scal

fference Equations

Volume 2008, Article ID 796851, 13 pages

doi:10.1155/2008/796851

Research Article

Multiple Positive Solutions in the Sense of

Distributions of Singular BVPs on Time Scales and

an Application to Emden-Fowler Equations

Ravi P Agarwal, 1 Victoria Otero-Espinar, 2 Kanishka Perera, 1

and Dolores R Vivero 2

1 Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA

2 Departamento de An´alise Matem´atica, Facultade de Matem´aticas, Universidade de Santiago

de Compostela, 15782 Santiago de Compostela, Galicia, Spain

Correspondence should be addressed to Ravi P Agarwal,agarwal@fit.edu

Received 21 April 2008; Accepted 17 August 2008

Recommended by Paul Eloe

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

Copyrightq 2008 Ravi P Agarwal et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

The Emden-Fowler equation,

uΔΔt  qt u α

σ t 0, t ∈ 0, 1T, 1.1 arises in the study of gas dynamics and fluids mechanics, and in the study of relativistic mechanics, nuclear physics, and chemically reacting systemsee, e.g., 1

therein for the continuous model The negative exponent Emden-Fowler equation α < 0 has been used in modeling non-Newtonian fluids such as coal slurries 2

interest lies in the existence of positive solutions We are interested in a broad class of singular problem that includes those related with1.1 and the more general equation

uΔΔt  qt u α

σ t gt, u σ t, t ∈ 0, 1T. 1.2 Recently, existence theory for positive solutions of second-order boundary value problems on time scales has received much attentionsee, e.g., 3 6 7 for the continuous case, and 8

In this paper, we consider the second-order dynamic equation with homogeneous Dirichlet boundary conditions:

P

⎧

⎪

⎨

⎪

⎩

−uΔΔt  Ft, u σ t, Δ-a.e t ∈D κo

,

u t > 0, t ∈ a, bT,

u a  0  ub,

1.3

where we say that a property holds forΔ-a.e t ∈ A ⊂ T or Δ-a.e on A ⊂ T, Δ-a.e., whenever there exists a set E ⊂ A with null Lebesgue Δ-measure such that this property holds for every

t ∈ A \ E, T is an arbitrary time scale, subindex T means intersection to T, a, b ∈ T are such that a < ρ T, D κ

T, D κ2

 a, ρ2

T, D o  a, bT,D κo  a, ρbT,

and F : D × 0, ∞ → R is an L1

Δ-Carath´eodory function on compact subintervals of0, ∞,

that is, it satisfies the following conditions

C i For every x ∈ 0, ∞, F·, x is Δ-measurable in D o

ii For Δ-a.e t ∈ D o , Ft, · ∈ C0, ∞.

Cc For every x1, x2∈ 0, ∞ with x1≤ x2, there exists m x1,x2 ∈ L1

ΔD o such that

F t, x ≤ m x1,x2 t for Δ-a.e t ∈ D o , x∈ x1, x2 . 1.4

Moreover, in order to use variational techniques and critical point theory, we will

assume that F satisfy the following condition.

PM For every x ∈ 0, ∞, function P F : D × 0, ∞ → R defined for Δ-a.e t ∈ D and all x ∈ 0, ∞, as

P F t, x :

x

0

satisfies that P F ·, x is Δ-measurable in D o

We consider the spaces

C10,rd

D κ : C1 rd



D κ

∩ C0D,

Cc,rd1 

D κ : C1 rd



D κ

where C1

rdD κ  is the set of all continuous functions on D such that they are Δ-differentiable

on D κ and theirΔ-derivatives are rd-continuous on D κ , C0D is the set of all continuous functions on D that vanish on the boundary of D, and CcD is the set of all continuous functions on D with compact support on a, bT We denote as·C D the norm in CD, that

is, the supremum norm

On the other hand, we consider the first-order Sobolev spaces

ΔD :v : D −→ R : v ∈ ACD, vΔ∈ L2

Δ



0,Δ D :v : D −→ R : v ∈ H1

ΔD, va  0  vb ,

1.7

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

t2

t1

f s Δs 

t1,t2 Tf sΔs for t1, t2∈ D, t1< t2, f ∈ L1

Δ



t1, t2

T



The set H is endowed with the structure of Hilbert space together with the inner

product·, · H : H × H → R given for every v, w ∈ H × H by

v, w H:vΔ, wΔ

L2

Δ:

b

a

vΔs · wΔs Δs; 1.9

we denote as·Hits induced norm

Moreover, we consider the sets

H 0,loc: H1

loc,Δ D ∩ C0D,

H c,loc: H1

where H1

loc,Δ D is the set of all functions such that their restriction to every closed subinterval

J of a, bTbelong to the Sobolev space H1

ΔJ.

We refer the reader to 9 11

and absolutely continuous functions on closed subintervals of an arbitrary time scale, and to 12

Definition 1.1 u is said to be a solution in the sense of distributions to P if u ∈ H 0,loc , u > 0

ona, bT, and equality

b

a

uΔs · ϕΔs − Fs, u σ s· ϕ σ s Δs  0 1.11

holds for all ϕ ∈ C1

c,rdD κ

From the density properties of the first-order Sobolev spaces proved in 9, Seccion 3.2

we deduce that if u is solution in the sense of distributions, then,1.11 holds for all ϕ ∈ H c,loc This paper is devoted to prove the existence of multiple positive solutions toP by

using perturbation and variational methods

This paper is organized as follows InSection 2, we deduce sufficient conditions for the existence of solutions in the sense of distributions to P Under certain hypotheses,

we approximate solutions in the sense of distributions to problem P by a sequence of

weak solutions to weak problems InSection 3, we derive some sufficient conditions for the existence of at least one or two positive solutions toP.

These results generalize those given in 7

on the whole interval0, 1 ∩ T and the authors assume that F ∈ C0, 1 × 0, ∞, R instead

of C and PM 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 toP by weak problems

In this section, we will deduce sufficient conditions for the existence of solutions in the sense

of distributions toP, where F  f  g and f, g : D × 0, ∞ → R satisfy C and PM, f

satisfiesCc, and g satisfies the following condition.

Cg For every p ∈ 0, ∞, there exists M p ∈ L1

ΔD o such that

g t, x ≤ M p t for Δ-a.e t ∈ D o , x 2.1

Under these hypotheses, we will be able to approximate solutions in the sense of distributions to problemP by a sequence of weak solutions to weak problems.

First of all, we enunciate a useful property of absolutely continuous functions on

Dwhose proof we omit because of its simplicity.

Lemma 2.1 If v ∈ ACD, then v± : max{± v, 0} ∈ ACD,



vΔ

− vΔ ·vΔ

≤ 0, v−Δ

 vΔ ·v−Δ

Δ-a.e on D o

We fix {ε j}j≥1 a sequence of positive numbers strictly decreasing to zero; for every

j ≥ 1, we define f j : D × 0, ∞ → R as

f j t, x  ft, max

x, ε j for everyt, x ∈ D × 0, ∞. 2.3

Note that f jsatisfiesC and C g; consider the following modified weak problem



P j



⎧

⎪

⎪

−uΔΔt  f j



t, u σ t gt, u σ t, Δ-a.e t ∈D κo

,

u a  0  ub.

2.4

Definition 2.2 u is said to be a weak solution to P j  if u ∈ H, u > 0 on a, bT, and equality

b

a

uΔs · ϕΔs −f j

s, u σ s gs, u σ s· ϕ σ s Δs  0 2.5

holds for all ϕ ∈ C1

0,rdD κ

u is said to be a weak lower solution to P j  if u ∈ H u > 0 on a, bT, and inequality

b

a

uΔs · ϕΔs −f j

s, u σ s gs, u σ s· ϕ σ s Δs ≤ 0 2.6

holds for all ϕ ∈ C1

0,rdD κ  such that ϕ ≥ 0 on D.

The concept of weak upper solution to P j 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 Definition 2.2are valid for all ϕ ∈ H and for all

ϕ ∈ H such that ϕ ≥ 0 on D, respectively.

By standard arguments, we can prove the following result

Proposition 2.3 Assume that f, g : D × 0, ∞ → R satisfy (C and (PM, f satisfies (C c , and

g satisfies (C g .

Then, if for some j ≥ 1 there exist u j and u j as a lower and an upper weak solution, respectively,

to P j  such that u j ≤ u j on D, then P j  has a weak solution u j ∈ u j , u j j ≤ v ≤

u j on D }.

Next, we will deduce the existence of one solution in the sense of distributions to

P from the existence of a sequence of weak solutions to P j In order to do this, we fix

{a k}k≥1, {b k}k≥1 ⊂ D two sequences such that {a k}k≥1⊂ a, a  b/2T is strictly decreasing

to a if a  σa, a k  a for all k ≥ 1 if a < σa and {b k}k≥1 ⊂ a  b/2, bT is strictly

increasing to b if ρb  b, b k  b for all k ≥ 1 if ρb < b We denote that D k : ak , b k T,

k ≥ 1 Moreover, we fix {δ k}k≥1a sequence of positive numbers strictly decreasing to zero such that

σ

a k



, ρ b k



T⊂ a  δ k , b − δ k



T, δ k≤ b − a

Proposition 2.4 Suppose that F  f  g and f, g : D × 0, ∞ → R satisfy (C and (PM, f

satisfies (C c , and g satisfies (C g .

Then, if for every j ≥ 1, u j ∈ H is a weak solution to P j  and

ν δ: inf

j≥1 min

T



0, b − a

2



M : sup

j≥1 max

then a subsequence of {u j}j≥1converges pointwise in D to a solution in the sense of distributions u1

to P.

Proof Let k ≥ 1 be arbitrary; we deduce, from 2.2, 2.7, 2.8, and 2.9, that there exists a

constant K k ≥ 0 such that for all j ≥ 1,

b k

a k



uΔj s2Δs uΔj

a k2

· μa k

uΔj

ρ b k

2

· μρ

b k



ρ b k

σ a kuΔj s ·

u j − ν δ k

Δ

s Δs

≤ K ku j ,

u j − ν δ k



H

2.10

Therefore, for all j ≥ 1 so large that ε j < ν δ1, as u jis a weak solution toP j, by taking

ϕ1: uj − ν δ1 ∈ H as the test function in 2.5, from 2.9, Cc and Cg, we can assert that

there exists l ∈ L1

ΔD o such that

b1

a1



uΔj s2Δs ≤ K1

b

a

F

s, u σ

j s· ϕ σ

1sΔs

≤ K1 M

b

a

l sΔs,

2.11

that is,{u j}j≥1is bounded in H1

ΔD1 and hence, there exists a subsequence {u1j}j≥1which

converges weakly in HΔ1D1 and strongly in CD1 to some u1∈ H1

ΔD1

For every k ≥ 1, by considering for each j ≥ 1 the weak solution to P k j u k j and

by repeating the previous construction, we obtain a sequence{u k1 j}j≥1 which converges

weakly in H1

ΔD k1 and strongly in CD k1 to some u k1 ∈ H1

ΔD k1 with {u k1 j}j≥1 ⊂

{u k j}j≥1 By definition, we know that for all k ≥ 1, u k1|D k  u k

Let u1: D → R be given by u1 : uk on D k for all k ≥ 1 and u1a : 0 : u1b so that

u1 > 0 on a, bT, u1 ∈ H1

loc,Δ D ∩ Ca, bT, u1is continuous in every isolated point of the

boundary of D, and {u k k}k≥1converges pointwise in D to u1

We will show that u1 ∈ C0D; we only have to prove that u1is continuous in every

dense point of the boundary of D Let 0 < ε < M be arbitrary, it follows fromCc and Cg

that there exist m ε ∈ L1

ΔD o  such that m ε ≥ 0 on D o and Ft, x ≤ m ε t for Δ-a.e t ∈ D oand

all x ε ∈ H be the weak solution to

−ϕΔΔ

ε t  m ε t, Δ-a.e t ∈D κo

, ϕ ε a  0  ϕ ε b; 2.12

we knowsee 4 ε > 0 on a, bT

For all k ≥ 1 so large that ε k k < ε, since u k k and ϕ εare weak solutions to some problems,

by taking ϕ2 u k k − ε − ϕ ε∈ H as the test function in their respective problems, we obtain



u k k , ϕ2



H

b

a

F

s, u σ k

k s· ϕ σ

2s Δs

≤

b

a

m ε s · ϕ σ

2s Δs ϕ ε , ϕ2



H;

2.13

thus,2.2 yields to

ϕ22

H≤u k k − ϕ ε , ϕ2



which implies that 0≤ u k k ≤ ε  ϕ ε on D and so 0 ≤ u1≤ ε  ϕ ε on D Thereby, the continuity

of ϕ ε in every dense point of the boundary of D and the arbitrariness of ε guarantee that

u1∈ C0D.

Finally, we will see that1.11 holds for every test function ϕ ∈ C1

c,rd D κ; fix one of them

For all k ≥ 1 so large that supp ϕ ⊂ a k , b kTand all j ≥ 1 so large that ε k j < ν δ k , as u k j

is a weak solution toP k j , by taking ϕ ∈ C1

c,rd D κ  ⊂ C1

0,rd D κ as the test function in 2.5 and bearing in mind2.7, we have

b k

a

uΔk

j s · ϕΔsΔs u k j , ϕ

H

b k

a

F

s, u σ k

j s· ϕ σ sΔs, 2.15

whence it follows, by taking limits, that

b k

a k



u kΔ

s · ϕΔs − Fs,

u kσ s· ϕ σ sΔs  0, 2.16

which is equivalent because u1|D k  u k and ϕ  0  ϕ σ on D o \ D o

kto b

a



uΔ1s · ϕΔs − Fs, u σ1s· ϕ σ sΔs  0, 2.17 and the proof is therefore complete

Propositions2.3and2.4lead to the following sufficient condition for the existence of

at least one solution in the sense of distributions to problemP.

Corollary 2.5 Let F  f  g be such that f, g : D × 0, ∞ → R satisfy (C and (PM, f satisfies

(C c , and g satisfies (C g .

Then, if for each j ≥ 1 there exist u j and u j a lower and an upper weak solution, respectively,

to P j  such that u j ≤ u j on D and

inf

j≥1 min

T

u j > 0 ∀δ ∈



0, b − a

2



j≥1

max

D u j < ∞, 2.18

then P has a solution in the sense of distributions u1.

Finally, fixed u1∈ H 0,locis a solution in the sense of distributions toP with F  f  g,

we will derive the existence of a second solution in the sense of distributions toPgreater than or equal to u1on D For every k≥ 1, consider the weak problem

 P k⎧⎨

⎩

−vΔΔt  Ft,

u1 vσ

t− Ft, u σ

1t, Δ-a.e t ∈D κ

k

o

,

v

a k

 0  vb k

For every k ≥ 1, consider H k : H1

0,Δ D k  as a subspace of H by defining it for every

v ∈ H k as v  0 on D \ D kand define the functionalΦk : H k ⊂ H → R for every v ∈ H kas

Φk v : 1

2v2

H−

b k

a k

G

s,

vσ

where function G : D × 0, ∞ → R is defined for Δ-a.e t ∈ D and all x ∈ 0, ∞ as

G t, x :

x

0



F

t, u σ1t  r− Ft, u σ1tdr. 2.21

As a consequence of Lemma 2.1, we deduce that every weak solution to  P k is

nonnegative on D k and by reasoning as in 4, Section 3 k is weakly lower semicontinuous,Φkis continuously differentiable in Hk , for every v, w ∈ H k,

Φ

k vw  v, w H−

b k

a k



F

s,

u1 vσ

s− Fs, u σ1s· w σ sΔs, 2.22

and weak solutions to P k match up to the critical points of Φk

Next, we will assume the following condition

NI For Δ-a.e t ∈ D o , ft, · is nonincreasing on 0, ∞.

Proposition 2.6 Suppose that F  f  g is such that f, g : D × 0, ∞ → R satisfy (C and (PM,

f satisfies (C c  and (NI, and g satisfies (C g .

If {v k}k≥1⊂ H, v k ∈ H k is a bounded sequence in H such that

inf

k≥1Φk



v k

k→∞Φ

k



v k

v ≥ 0 in D and u2: u1 v is a solution in the sense of distributions to P.

strongly in C0D to some v ∈ H.

For every k≥ 1, by 2.2, we obtain

v−

k

H≤Φ

k



v k

which implies, from2.23, that v ≥ 0 on D and so u2: u1 v > 0 on a, bT

In order to show that u2 : u1 v ∈ H 0,loc is a solution in the sense of distributions

toP, fix ϕ ∈ C1

c,rd D k  arbitrary and choose k ≥ 1 so large that supp ϕ ⊂ a k , b kT, bearing

in mind that u1 is a solution in the sense of distributions toP, and the pass to the limit in

2.22 with v  v k and w  ϕ yields to

0

b

a

vΔs · ϕΔs −F

s,

u1  vσ s− Fs, u σ1s· ϕ σ s Δs



b

a

uΔ2s · ϕΔs − Fs, u σ2s· ϕ σ s Δs;

2.25

thus, u2is a solution in the sense of distributions toP.

Finally, we will see that v is not the trivial function; suppose that v  0 on D Condition

NIensures that function G defined in 2.21 satisfies for every k ≥ 1 and Δ-a.e s ∈ D o,

G

s,

v kσ

s≥f

s,

u1  v

k

σ

s− fs, u σ1s·vkσ

s



v

kσ s

0



g

s, u σ

1s  r− gs, u σ

1sdr,

2.26

so that, by2.20 and 2.22, we have, for every k ≥ 1,

Φk



v k

≤ 1

2v k2

H−v k , v k

H Φ

k



v k

vk

− b

a



g

s,

u1  v

k

σ

s− gs, u σ1s·vkσ

sΔs



b

a

v

kσ s

0



g

s, u σ1s  r− gs, u σ1sdr



Δs;

2.27

moreover, as we know that vk ≤ p on D for some p > 0, it follows from Cg that there exists

ΔD o such that

Φk



v k

≤ 1 2

v−

k2

H−v

k2

H



 Φ

k



v k

v k

 2

b

a

m s ·vkσ sΔs

≤ 1

2v−

k2

HΦ

k



v k

H∗k·v

k

H 2

b

a

m s ·vkσ

sΔs,

2.28

and hence, since{v

k}k≥1is bounded in H and converges pointwise in D to the trivial function

contradicts the first relation in2.23 Therefore, v 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 toP where F  f  g0 ηg1, η ≥ 0 is a small parameter, and f, g0, g1: D × 0, ∞ → R satisfy C,

PM as well as the following conditions

H1 There exists a constant x0 ∈ 0, ∞ and a nontrivial function f0 ∈ L1

ΔD o such

that f0 ≥ 0 Δ-a.e on D oand

f t, x ≥ f0t, g0t, x, g1t, x ≥ 0 for Δ-a.e t ∈ D o , x∈0, x0 . 3.1

H2 For every p ∈ 0, ∞, there exist m p ∈ L1

ΔD o  and K p≥ 0 such that

f t, x ≤ m p t for Δ-a.e t ∈ D o , x ∈ p, ∞,

g1 t, x ≤ K p forΔ-a.e t ∈ D o , x 3.2

H3 There are m0∈ L2

ΔD o such that

g0t, x ≤ λx  m0t for Δ-a.e t ∈ D o , x ∈ 0, ∞, 3.3

for some λ < λ1, where λ1is the smallest positive eigenvalue of problem

−uΔΔt  λu σ t, t ∈ D κ2

,

3.1 Existence of one solution Uniqueness

Theorem 3.1 Suppose that f, g0, g1 : D × 0, ∞ → R satisfy (C, (PM, and (H1–H3 Then,

there exists a η0 > 0 such that for every η ∈ 0, η0, problem P with F  f  g0 ηg1has a solution

in the sense of distributions u1.

Cg We will show that there exists a η0 > 0 such that for every η ∈ 0, η0, hypotheses in Corollary 2.5are satisfied

Let x0and f0be given inH1, we know, from 4, Proposition 2.7

ε

−uΔΔt  εf0t, Δ-a.e t ∈D κo

, u a  0  ub, 3.5

satisfies that u > 0 on a, bTand u ≤ x0on D.

Let j ≥ 1 be so large that ε j < x0, we obtain, byH1, that

−uΔΔt ≤ f0t ≤ f j



t, u σ t gt, u σ t, Δ-a.e t ∈ D o , 3.6

whence it follows that u is a weak lower solution to P j

As a consequence ofC, PM, and H1–H3, by reasoning as in 4, Theorem 4.2

we deduce that problem

−uΔΔt  f j



t, u σ t g0



t, u σ t 1, Δ-a.e t ∈D κo

,

u t > 0, t ∈ a, bT,

u a  0  ub

3.7

has some weak solution u j ∈ H which, fromLemma 2.1andH1, satisfies that u ≤ u j on

function, we know from2.2, H2, and H3 that there exist m x0 ∈ L2

ΔD o such that

ϕ j2

H≤u j − x0, ϕ j

H



b

a



f j

s, u σ j s g0



s, u σ j s 1· ϕ σ

j sΔs

≤ b

a



λu σ j s  m x0s  m0s  1· ϕ σ

j sΔs;

3.8

so that, it follows from the fact that the immersion from H into C0D is compact, see 9, Proposition 3.7 10, Corollary 3.2 1that{ϕ j}j≥1is

bounded in H and, hence, {u j}j≥1is bounded in C0D Thereby, condition H2 allows to

assert that there exists η0≥ 0, such that for all η ∈ 0, η0

−uΔΔj t ≥ f j



t, u σ j t g0



t, u σ j t ηg1



t, u σ j t, Δ-a.e t ∈ D o , 3.9

holds, which implies that u jis a weak upper solution toP j

Therefore, for every j ≥ 1 so large, we have a lower and an upper solution to P j, respectively, such that2.2 is satisfied and so,Corollary 2.5guarantees that problemP has

at least one solution in the sense of distributions u1

Theorem 3.2 If f : D × 0, ∞ → R satisfies (C, (C c , and (NI, then, P with F  f has at

most one solution in the sense of distributions.

Proof Suppose that P has two solutions in the sense of distributions u1, u2∈ H 0,loc Let ε > 0

be arbitrary, take ϕ  u1− u2− ε ∈ H c,locas the test function in1.11, by 2.2 and NI,

we have

ϕ2

H≤u1− u2− ε, ϕH

b

a



f

s, u σ1s− fs, u σ2s· ϕ σ sΔs ≤ 0, 3.10

thus, u1 ≤ u2 ε on D The arbitrariness of ε leads to u1 ≤ u2on D and by interchanging u1

and u2, we conclude that u1 u2on D.

Corollary 3.3 If f : D × 0, ∞ → R satisfies (C, (PM, (NI, and (H1-(H2 with g0 0  g1,

then P with F  f has a unique solution in the sense of distributions.

we approximate solutions in the sense of distributions to problem... 2.17 and the proof is therefore complete

Propositions2. 3and2 .4lead to the following sufficient condition for the existence of

at least one solution in the sense of distributions to. .. refer the reader to 9 11

and absolutely continuous functions on closed subintervals of an arbitrary time scale, and to 12

Definition 1.1 u is said to be a solution in the sense of