# Positive Solutions for a Class of Nonlinear Boundary Value

## Transcript Of Positive Solutions for a Class of Nonlinear Boundary Value

Journal of Mathematical Analysis and Applications 236, 94᎐124 Ž1999. Article ID jmaa.1999.6439, available online at http:rrwww.idealibrary.com on

Positive Solutions for a Class of Nonlinear Boundary Value Problems with Neumann᎐Robin Boundary Conditions

w metadata, citation and similar papers at coVre..aAc.nukuradha

2401 W. Spring Creek Pkwy, Plano, Texas 75023 E-mail: [email protected]

and

C. Maya and R. Shivaji

Mississippi State Uni¨ersity, Mississippi State, Mississippi 39762 E-mail: [email protected], [email protected]

Submitted by Zhi¨ko S. Athanasso¨ Received March 4, 1998

We consider the two point boundary value problem

yuЉ Ž x . s f Ž uŽ x .. ; 0 - x - 1

uЈŽ0. s 0;

uЈŽ1. q ␣ uŽ1. s 0

where ) 0 and ␣ ) 0 are parameters, and f g C 2w0, 1x. We discuss the existence of nonnegative solutions for superlinear nonlinearities by developing a quadrature method. We study the positone Ž f Ž0. ) 0. case as well as the semipositone Ž f Ž0. - 0. case, and note a drastic difference in the respective bifurcation diagrams for positive solutions. ᮊ 1999 Academic Press

1. INTRODUCTION

Consider the nonlinear boundary value problem

yuЉ Ž x . s f Ž uŽ x . . ; 0 - x - 1 uЈŽ0. s 0;

uЈŽ1. q ␣ uŽ1. s 0

94 0022-247Xr99 $30.00

Copyright ᮊ 1999 by Academic Press All rights of reproduction in any form reserved.

Ž 1 .1. Ž 1 .2. Ž 1 .3.

POSITIVE SOLUTIONS

95

where ) 0 and ␣ ) 0 are parameters and f g C 2w0, 1x. For positone problems, since f Ž s. ) 0 for s G 0, for the case ␣ s 0 ŽuЈŽ0. s 0 s uЈŽ1.., one can easily show that Ž1.1.᎐Ž1.3. have no nonnegative solutions. For semipositone problems Ž f Ž0. - 0., existence and multiplicity results have been established for the case ␣ s 0 in w6x. In this paper, we study Ž1.1.᎐Ž1.3. for ␣ ) 0 for the positone as well as the semipositone problems. We establish our existence results by building a quadrature method. See also w1᎐5x where quadrature methods have been used to study other types of two point boundary value problems. We extend the quadrature mehtod used in w6x for Neumann boundary conditions to Ž1.2.᎐Ž1.3. with ␣ ) 0. Note that the quadrature method for ␣ ) 0 does not generate an explicit expression for describing the branches of solutions. But by analyz-

ing the implicit relationships, we prove our results. We establish the existence of nonnegative solutions for the case ␣ ) 0 and f Ž0. ) 0 in Section 2, and for the case ␣ ) 0 and f Ž0. - 0 in Section 3. One may analyze these relationships further to study uniqueness and multiplicity

results.

2. EXISTENCE RESULTS WHEN f Ž0. ) 0

Throughout this section, we will assume the following:

f Ž s. ) 0 for s G 0,

fŽ s.

lim

s ϱ,

sªϱ s

fЈŽ s. G 0, fЉ Ž s. ) 0,

Hs

F Ž s. s f Ž t . dt.

0

It is easy to see that all nonnegative solutions are positive in w0, 1x. We will

now discuss the quadrature technique for this case. Assume that uŽ x. is a positive solution of Ž1.1.᎐Ž1.3. with uŽ0. s . Note that since f Ž s. ) 0 for s G 0, uŽ x. is concave down and cannot have a critical point in Ž0, 1x. So uŽ x. has precisely the form shown in Fig. 2.1.

Multiplying Ž1.1. throughout by uЈŽ x., we have yuЉ Ž x.uЈŽ x. s f ŽuŽ x..uЈŽ x. and integrating the above equation, we obtain

uЈŽ x. 2 y 2 s FŽ uŽ x.. q C.

Since uŽ0. s , C s yFŽuŽ0.. s yFŽ . and therefore

uЈŽ x. 2 y 2 s F Ž uŽ x . . y F Ž . ; x g Ž0, 1. Ž2.1.

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ANURADHA ET AL.

FIGURE 2.1

and

uЈŽ x . s y'2 'F Ž . y F Ž uŽ x . . ; x g Ž0, 1. .

Ž 2 .2.

Integrating Ž2.2. on Ž0, x., we have

HuŽ x. ds s y'2 x ; ' uŽ0. F Ž . y F Ž s.

x g Ž0, 1. .

Ž 2 .3.

Let

uЈŽ1. s ym where

m ) 0. Then

uŽ1. s

m ␣

g

Ž0,

..

Substituting

xs1

in Ž2.3., we obtain

1

ds

H ' s

' '2 mr␣ F Ž . y F Ž s.

Ž 2 .4.

and substituting x s 1 in Ž2.2., we get

'm

' s

.

'2 F Ž . y F Ž mr␣ .

Ž 2 .5.

Combining Ž2.4. and Ž2.5., for such a solution to exist, there must exist an m such that

ds

m

Hs

.

' ' mr␣ F Ž . y F Ž s.

F Ž . y F Ž mr␣ .

Ž 2 .6.

We first investigate whether such an m exists.

POSITIVE SOLUTIONS

97

For m g Ž0, ␣ ., define GŽ m. s Hm r␣

ds

. Then GŽ0. s

'F Ž . y F Ž s.

' ' H0 ds

) 0, GŽ ␣ . s 0, and GЈŽ m. s

y1

. Hence,

FŽ . y FŽs.

␣ F Ž . y F Žmr␣ .

for a given ␣ g Ž0, ϱ. and g Ž0, ϱ., GŽ m. is a decreasing function of m

Žsee Fig. 2.2.. Let HŽ m. s

m

; then HŽ0. s 0, HŽ m. ª ϱ as

'F Ž . y F Ž mr␣ .

m ª ␣ , and

2 ␣ F Ž . y F Ž mr␣ . q mf Ž mr␣ .

HЈŽ m. s

2 ␣ F Ž . y F Ž mr␣ . 3r2

) 0.

Hence, for a given ␣ g Ž0, ϱ. and g Ž0, ϱ., HŽ m. is an increasing function of m Žsee Fig. 2.2.. Thus, given ␣ g Ž0, ϱ. and g Ž0, ϱ., clearly there exists a unique m s m*Ž ␣ , . g Ž0, ␣ . such that GŽ m*. s HŽ m*..

Now by back-tracking, we can prove the existence of a positive solution uŽ x. to Ž1.1.᎐Ž1.3. as described in Theorem 2.1

THEOREM 2.1. Gi¨en ␣ g Ž0, ϱ. and g Ž0, ϱ., there exists a unique m s m*Ž ␣ , . g Ž0, ␣ . such that

ds

m

Hs

' ' mr␣ F Ž . y F Ž s.

F Ž . y F Ž mr␣ .

for m s m*. Also, there exists a unique s Ž , m*. gi¨en by either

1

ds

H ' s

' '2 mr␣ F Ž . y F Ž s.

FIGURE 2.2

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ANURADHA ET AL.

or

'm

' s '2 F Ž . y F Ž mr␣ .

for which Ž1.1.᎐Ž1.3. has a unique positi¨e solution uŽ x. as defined by

HuŽ x.

ds

s y'2 x ;

' uŽ0. F Ž . y F Ž s.

x g Ž0, 1.

with uŽ0. s and uЈŽ1. s ym*.

COROLLARY 2.2. Gi¨en ␣ g Ž0, ϱ., the bifurcation diagram Ž , . of the positi¨e solutions of Ž1.1.᎐Ž1.3. is described by either

1

ds

H ' s

' '2 mr␣ F Ž . y F Ž s.

or

'm

' s '2 F Ž . y F Ž mr␣ .

for m s m*.

Since m* g Ž0, ␣ ., we have 0 - Ž , m*. - Ž . where ' Ž . s

1 H0 ds . Note that s Ž . describes the bifurcation diagram

'2 'F Ž . y F Ž s.

of positive solutions of the boundary value problem

yuЉ Ž x . s f Ž uŽ x . . ; uЈŽ0. s 0 s uŽ1.

0 - x - 1 Žy1 - x - 1. Ž uŽy1. . s 0 suŽ1. . .

Since lim sªϱ f Žss. s ϱ it follows that lim ªϱ Ž . s 0. Hence, given ) 0 and ␣ g Ž0, ϱ., our result on the existence of a positive solution for the positone case is described via the Ž , . bifurcation diagram Žsee Fig. 2.3. and Theorem 2.3.

THEOREM 2.3. Gi¨en g Ž0, Ž , m*.. and ␣ g Ž0, ϱ., Ž1.1.᎐Ž1.3. ha¨e at least one positi¨e solution with no interior critical points.

POSITIVE SOLUTIONS

99

FIGURE 2.3

3. EXISTENCE RESULTS WHEN f Ž0. - 0 Throughout this section we will assume the following: ᭚ , ) 0 such that f Ž s. - 0 on w0,  ., f Ž  . s 0, f ЈŽ s. G 0, f Љ Ž s. ) 0, lim sªϱ f Žss. s ϱ, and FŽ . s 0 where FŽ s. s H0s f Žt. dt. Here also we will characterize our nonnegative solutions by the value of the solution at x s 0, which we again denote by . Recall that in the positone case for each given ) 0 there exist a unique for which a positive solution exists with no critical points in Ž0, 1x. Further, no solution with critical points in Ž0, 1x exists for any . However, for the semipositone case, this is not always the case. It follows that given n s 0, 1, 2, . . . , there exist ranges of g w0, x Ždepending on n. where there are exactly two

FIGURE 3.1

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ANURADHA ET AL.

values of , say, 1Ž , n. and 2Ž , n. for which Ž1.1.᎐Ž1.3. have nonnegative solutions with n interior critical points. We denote these solutions by

un, 1Ž , 1, ␣ . and un, 2Ž , 2 , ␣ . respectively. Further, there are values of g Ž0, . for which such a value of is unique, and ranges of g w0, x for which such a solution will not exist for any .

We will also see that if f Ä0, 4 then necessarily uЈŽ1. - 0, while if g Ä0, 4 then for each n s 0, 1, 2, . . . , there exists a solution with uЈŽ1. - 0 at s 1 and a second solution with uЈŽ1. s 0 at s 2.

Now, for ) , one cannot find a for which Ž1.1.᎐Ž1.3. have positive solutions with interior critical points. However, for ) , one can find a unique s 1Ž . such that Ž1.1.᎐Ž1.3. have a positive solution with no interior critical point, which we will denote by u0, 1Ž , 1, ␣ ..

In Section 3.1 we will discuss the case when ) . In Section 3.2 we discuss the case when g Ž , .. In Section 3.3 we will study the case when g Ž0,  .. In Section 3.4 we will discuss the case when s and s 0. Finally, in Section 3.5 we will provide a bifurcation diagram describ-

ing our results.

3.1. Existence Results when )

Let uŽ x. be a nonnegative solution of Ž1.1.᎐Ž1.3. with uŽ0. s g Ž , ϱ.. Since )  and f ) 0 for u ) , u must be initially concave down. Further, there does not exist 0 F r - such that FŽ . s FŽ r .. Therefore uŽ x. cannot have an interior critical point in Ž0, 1x wby Ž3.1.1. belowx and so the solution must be of the form as in Fig. 3.2. Again we notice that all nonnegative solutions are positive in w0, 1x.

FIGURE 3.2

POSITIVE SOLUTIONS

101

Then building a quadrature method as was done in section 2, we get

uЈŽ x. 2

y 2

s FŽ uŽ x.. y FŽ . ;

x g Ž0, 1. Ž3.1.1.

and

uЈŽ x . s y'2 'F Ž . y F Ž uŽ x . . ; x g Ž0, 1. . Ž3.1.2.

Integrating Ž3.1.2. on Ž0, x., we have

HuŽ x.

ds

s y'2 x ;

' uŽ0. F Ž . y F Ž s.

x g Ž0, 1. .

Ž 3 .1 .3.

Let uЈŽ1. s ym where m ) 0. Then uŽ1. s mr␣ g Ž0, .. Substituting x s 1 in Ž3.1.3., we obtain

1

ds

H ' s

' '2 mr␣ F Ž . y F Ž s.

Ž 3 .1 .4.

and substituting x s 1 in Ž3.1.2., we get

'm

' s

.

'2 F Ž . y F Ž mr␣ .

Ž 3 .1 .5.

Combining Ž3.1.4. and Ž3.1.5., for such a solution to exist, there must exist an m such that

ds

m

Hs

.

' ' mr␣ F Ž . y F Ž s.

F Ž . y F Ž mr␣ .

Ž 3 .1 .6.

We now investigate whether such an m exists. Given g Ž , ϱ., if we define

ds

H GŽ m. s

' mr␣ F Ž . y F Ž s.

'm

and H Ž m. s

,

F Ž . y F Ž mr␣ .

then GŽ m. is a decreasing function of m. Further, HŽ0. s 0, HŽ m. ª ϱ as m ª ␣ and

2 ␣ F Ž . y F Ž mr␣ . q mf Ž mr␣ . HЈŽ m. s 2 ␣ F Ž . y F Ž mr␣ . 3r2 .

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Note that HЈŽ m. ) 0 for m G ␣. For m g Ž0, ␣ ., define kŽ m. s

2 ␣ w FŽ . y FŽ mr␣ .x q mfŽ mr␣ .. Then kŽ0. s 2 ␣ FŽ . ) 0, kЈŽ m. s

yfŽ mr␣ . q

m ␣

f ЈŽ mr␣

.

)

0

since

f Ž s. - 0 on Ž0,  . and

f ЈŽ s. G 0. Thus,

kŽ m. ) 0 on Ž0, ␣ . and hence HŽ m. is an increasing function of m. Thus

GŽ m. and HŽ m. has the shape as described in Fig. 3.3. Therefore there exists a unique mU0, 1 s mU0, 1Ž ␣ , . such that GŽ mU0, 1. s HŽ mU0, 1..

Now by back-tracking, one can prove the existence of positive solution

to Ž1.1.᎐Ž1.3. as described in Theorem 3.1.1.

THEOREM 3.1.1. Gi¨en ␣ g Ž0, ϱ. and g Ž , ϱ., there exists a unique mU0, 1 s mU0, 1Ž ␣ , . such that

ds

m

Hs

' ' mr␣ F Ž . y F Ž s.

F Ž . y F Ž mr␣ .

for m s mU0, 1. Also, there exists a unique s 0, 1Ž , mU0, 1. gi¨en by either

1

ds

H ' s

' '2 mr␣ F Ž . y F Ž s.

or

'm

' s '2 F Ž . y F Ž mr␣ .

for which Ž1.1.᎐Ž1.3. ha¨e a unique positi¨e solution uŽ x. as defined by

HuŽ x.

ds

s y'2 x ;

' uŽ0. F Ž . y F Ž s.

with uŽ0. s and uЈŽ1. s ymU0, 1.

x g Ž0, 1.

FIGURE 3.3

POSITIVE SOLUTIONS

103

COROLLARY 3.1.2. Gi¨en ␣ g Ž0, ϱ., the bifurcation diagram Ž , . of the positi¨e solutions of Ž1.1.᎐Ž1.3. with uŽ0. s ) is described by either

1

ds

H ' s

' '2 mr␣ F Ž . y F Ž s.

or

'm

' s '2 F Ž . y F Ž mr␣ .

for m s mU0, 1.

Since mU0, 1 g Ž0, ␣ ., we

' ' Ž . s 1 H0

ds

'2

FŽ . y FŽs.

solutions of the problem

have 0 - 0, 1Ž , mU0, 1. - Ž . where s describes the bifurcation diagram of positive

yuЉ Ž u. s f Ž uŽ x . . ; uЈŽ0. s 0 s uŽ1.

0 - x - 1 Žy1 - x - 1. Ž uŽy1. . s 0 suŽ1. . .

3.2. Existence Results when g w , .

We first note that there cannot be a nonnegative solution of Ž1.1.᎐Ž1.3. with uŽ0. s  for any . This follows from the fact that the unique solution of

yuЉ Ž x . s f Ž uŽ x . . ; 0 - x - 1

uЈŽ0. s 0, uŽ0. s 

is uŽ x. '  Žsince f Ž  . s 0. while uŽ x. '  does not satisfy Ž1.3.. Now assume that uŽ x. is a positive solution of Ž1.1.᎐Ž1.3. with uŽ0. s

g Ž , .. Since ) , u must be initially concave down. Also by Ž1.3. and the fact that s , we must have uЈŽ1. s y␣ uŽ1. - 0. ŽFor if uŽ1. s 0 then uЈŽ1. s 0 and hence by Ž3.2.1.. Žbelow. we get FŽ . s FŽ0. s 0, which is a contradiction to the fact that FŽ . / 0.. Thus u must have either none or an even number of critical points.

Now, in general we consider a solution with 2 n interior critical points where n s 0, 1, 2, . . . Žsee Fig. 3.4.. Note that since Ž1.1. is autonomous, every solution of Ž1.1.᎐Ž1.3. is symmetric about each of its critical points. Therefore it is enough to study solutions between w0, x0 x and w2 nx0, 1x, where x0 is the first interior critical point. Let uŽ0. s uŽ2 x0. s иии s uŽ2 nx0. s g Ž , .. Let r g Ž0,  . be the unique number such that FŽ . s FŽ r . Žsee Fig. 3.5.. Then uŽ x0. s uŽ3 x0. s иии s uŽŽ2 n y 1. x0.

Positive Solutions for a Class of Nonlinear Boundary Value Problems with Neumann᎐Robin Boundary Conditions

w metadata, citation and similar papers at coVre..aAc.nukuradha

2401 W. Spring Creek Pkwy, Plano, Texas 75023 E-mail: [email protected]

and

C. Maya and R. Shivaji

Mississippi State Uni¨ersity, Mississippi State, Mississippi 39762 E-mail: [email protected], [email protected]

Submitted by Zhi¨ko S. Athanasso¨ Received March 4, 1998

We consider the two point boundary value problem

yuЉ Ž x . s f Ž uŽ x .. ; 0 - x - 1

uЈŽ0. s 0;

uЈŽ1. q ␣ uŽ1. s 0

where ) 0 and ␣ ) 0 are parameters, and f g C 2w0, 1x. We discuss the existence of nonnegative solutions for superlinear nonlinearities by developing a quadrature method. We study the positone Ž f Ž0. ) 0. case as well as the semipositone Ž f Ž0. - 0. case, and note a drastic difference in the respective bifurcation diagrams for positive solutions. ᮊ 1999 Academic Press

1. INTRODUCTION

Consider the nonlinear boundary value problem

yuЉ Ž x . s f Ž uŽ x . . ; 0 - x - 1 uЈŽ0. s 0;

uЈŽ1. q ␣ uŽ1. s 0

94 0022-247Xr99 $30.00

Copyright ᮊ 1999 by Academic Press All rights of reproduction in any form reserved.

Ž 1 .1. Ž 1 .2. Ž 1 .3.

POSITIVE SOLUTIONS

95

where ) 0 and ␣ ) 0 are parameters and f g C 2w0, 1x. For positone problems, since f Ž s. ) 0 for s G 0, for the case ␣ s 0 ŽuЈŽ0. s 0 s uЈŽ1.., one can easily show that Ž1.1.᎐Ž1.3. have no nonnegative solutions. For semipositone problems Ž f Ž0. - 0., existence and multiplicity results have been established for the case ␣ s 0 in w6x. In this paper, we study Ž1.1.᎐Ž1.3. for ␣ ) 0 for the positone as well as the semipositone problems. We establish our existence results by building a quadrature method. See also w1᎐5x where quadrature methods have been used to study other types of two point boundary value problems. We extend the quadrature mehtod used in w6x for Neumann boundary conditions to Ž1.2.᎐Ž1.3. with ␣ ) 0. Note that the quadrature method for ␣ ) 0 does not generate an explicit expression for describing the branches of solutions. But by analyz-

ing the implicit relationships, we prove our results. We establish the existence of nonnegative solutions for the case ␣ ) 0 and f Ž0. ) 0 in Section 2, and for the case ␣ ) 0 and f Ž0. - 0 in Section 3. One may analyze these relationships further to study uniqueness and multiplicity

results.

2. EXISTENCE RESULTS WHEN f Ž0. ) 0

Throughout this section, we will assume the following:

f Ž s. ) 0 for s G 0,

fŽ s.

lim

s ϱ,

sªϱ s

fЈŽ s. G 0, fЉ Ž s. ) 0,

Hs

F Ž s. s f Ž t . dt.

0

It is easy to see that all nonnegative solutions are positive in w0, 1x. We will

now discuss the quadrature technique for this case. Assume that uŽ x. is a positive solution of Ž1.1.᎐Ž1.3. with uŽ0. s . Note that since f Ž s. ) 0 for s G 0, uŽ x. is concave down and cannot have a critical point in Ž0, 1x. So uŽ x. has precisely the form shown in Fig. 2.1.

Multiplying Ž1.1. throughout by uЈŽ x., we have yuЉ Ž x.uЈŽ x. s f ŽuŽ x..uЈŽ x. and integrating the above equation, we obtain

uЈŽ x. 2 y 2 s FŽ uŽ x.. q C.

Since uŽ0. s , C s yFŽuŽ0.. s yFŽ . and therefore

uЈŽ x. 2 y 2 s F Ž uŽ x . . y F Ž . ; x g Ž0, 1. Ž2.1.

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ANURADHA ET AL.

FIGURE 2.1

and

uЈŽ x . s y'2 'F Ž . y F Ž uŽ x . . ; x g Ž0, 1. .

Ž 2 .2.

Integrating Ž2.2. on Ž0, x., we have

HuŽ x. ds s y'2 x ; ' uŽ0. F Ž . y F Ž s.

x g Ž0, 1. .

Ž 2 .3.

Let

uЈŽ1. s ym where

m ) 0. Then

uŽ1. s

m ␣

g

Ž0,

..

Substituting

xs1

in Ž2.3., we obtain

1

ds

H ' s

' '2 mr␣ F Ž . y F Ž s.

Ž 2 .4.

and substituting x s 1 in Ž2.2., we get

'm

' s

.

'2 F Ž . y F Ž mr␣ .

Ž 2 .5.

Combining Ž2.4. and Ž2.5., for such a solution to exist, there must exist an m such that

ds

m

Hs

.

' ' mr␣ F Ž . y F Ž s.

F Ž . y F Ž mr␣ .

Ž 2 .6.

We first investigate whether such an m exists.

POSITIVE SOLUTIONS

97

For m g Ž0, ␣ ., define GŽ m. s Hm r␣

ds

. Then GŽ0. s

'F Ž . y F Ž s.

' ' H0 ds

) 0, GŽ ␣ . s 0, and GЈŽ m. s

y1

. Hence,

FŽ . y FŽs.

␣ F Ž . y F Žmr␣ .

for a given ␣ g Ž0, ϱ. and g Ž0, ϱ., GŽ m. is a decreasing function of m

Žsee Fig. 2.2.. Let HŽ m. s

m

; then HŽ0. s 0, HŽ m. ª ϱ as

'F Ž . y F Ž mr␣ .

m ª ␣ , and

2 ␣ F Ž . y F Ž mr␣ . q mf Ž mr␣ .

HЈŽ m. s

2 ␣ F Ž . y F Ž mr␣ . 3r2

) 0.

Hence, for a given ␣ g Ž0, ϱ. and g Ž0, ϱ., HŽ m. is an increasing function of m Žsee Fig. 2.2.. Thus, given ␣ g Ž0, ϱ. and g Ž0, ϱ., clearly there exists a unique m s m*Ž ␣ , . g Ž0, ␣ . such that GŽ m*. s HŽ m*..

Now by back-tracking, we can prove the existence of a positive solution uŽ x. to Ž1.1.᎐Ž1.3. as described in Theorem 2.1

THEOREM 2.1. Gi¨en ␣ g Ž0, ϱ. and g Ž0, ϱ., there exists a unique m s m*Ž ␣ , . g Ž0, ␣ . such that

ds

m

Hs

' ' mr␣ F Ž . y F Ž s.

F Ž . y F Ž mr␣ .

for m s m*. Also, there exists a unique s Ž , m*. gi¨en by either

1

ds

H ' s

' '2 mr␣ F Ž . y F Ž s.

FIGURE 2.2

98

ANURADHA ET AL.

or

'm

' s '2 F Ž . y F Ž mr␣ .

for which Ž1.1.᎐Ž1.3. has a unique positi¨e solution uŽ x. as defined by

HuŽ x.

ds

s y'2 x ;

' uŽ0. F Ž . y F Ž s.

x g Ž0, 1.

with uŽ0. s and uЈŽ1. s ym*.

COROLLARY 2.2. Gi¨en ␣ g Ž0, ϱ., the bifurcation diagram Ž , . of the positi¨e solutions of Ž1.1.᎐Ž1.3. is described by either

1

ds

H ' s

' '2 mr␣ F Ž . y F Ž s.

or

'm

' s '2 F Ž . y F Ž mr␣ .

for m s m*.

Since m* g Ž0, ␣ ., we have 0 - Ž , m*. - Ž . where ' Ž . s

1 H0 ds . Note that s Ž . describes the bifurcation diagram

'2 'F Ž . y F Ž s.

of positive solutions of the boundary value problem

yuЉ Ž x . s f Ž uŽ x . . ; uЈŽ0. s 0 s uŽ1.

0 - x - 1 Žy1 - x - 1. Ž uŽy1. . s 0 suŽ1. . .

Since lim sªϱ f Žss. s ϱ it follows that lim ªϱ Ž . s 0. Hence, given ) 0 and ␣ g Ž0, ϱ., our result on the existence of a positive solution for the positone case is described via the Ž , . bifurcation diagram Žsee Fig. 2.3. and Theorem 2.3.

THEOREM 2.3. Gi¨en g Ž0, Ž , m*.. and ␣ g Ž0, ϱ., Ž1.1.᎐Ž1.3. ha¨e at least one positi¨e solution with no interior critical points.

POSITIVE SOLUTIONS

99

FIGURE 2.3

3. EXISTENCE RESULTS WHEN f Ž0. - 0 Throughout this section we will assume the following: ᭚ , ) 0 such that f Ž s. - 0 on w0,  ., f Ž  . s 0, f ЈŽ s. G 0, f Љ Ž s. ) 0, lim sªϱ f Žss. s ϱ, and FŽ . s 0 where FŽ s. s H0s f Žt. dt. Here also we will characterize our nonnegative solutions by the value of the solution at x s 0, which we again denote by . Recall that in the positone case for each given ) 0 there exist a unique for which a positive solution exists with no critical points in Ž0, 1x. Further, no solution with critical points in Ž0, 1x exists for any . However, for the semipositone case, this is not always the case. It follows that given n s 0, 1, 2, . . . , there exist ranges of g w0, x Ždepending on n. where there are exactly two

FIGURE 3.1

100

ANURADHA ET AL.

values of , say, 1Ž , n. and 2Ž , n. for which Ž1.1.᎐Ž1.3. have nonnegative solutions with n interior critical points. We denote these solutions by

un, 1Ž , 1, ␣ . and un, 2Ž , 2 , ␣ . respectively. Further, there are values of g Ž0, . for which such a value of is unique, and ranges of g w0, x for which such a solution will not exist for any .

We will also see that if f Ä0, 4 then necessarily uЈŽ1. - 0, while if g Ä0, 4 then for each n s 0, 1, 2, . . . , there exists a solution with uЈŽ1. - 0 at s 1 and a second solution with uЈŽ1. s 0 at s 2.

Now, for ) , one cannot find a for which Ž1.1.᎐Ž1.3. have positive solutions with interior critical points. However, for ) , one can find a unique s 1Ž . such that Ž1.1.᎐Ž1.3. have a positive solution with no interior critical point, which we will denote by u0, 1Ž , 1, ␣ ..

In Section 3.1 we will discuss the case when ) . In Section 3.2 we discuss the case when g Ž , .. In Section 3.3 we will study the case when g Ž0,  .. In Section 3.4 we will discuss the case when s and s 0. Finally, in Section 3.5 we will provide a bifurcation diagram describ-

ing our results.

3.1. Existence Results when )

Let uŽ x. be a nonnegative solution of Ž1.1.᎐Ž1.3. with uŽ0. s g Ž , ϱ.. Since )  and f ) 0 for u ) , u must be initially concave down. Further, there does not exist 0 F r - such that FŽ . s FŽ r .. Therefore uŽ x. cannot have an interior critical point in Ž0, 1x wby Ž3.1.1. belowx and so the solution must be of the form as in Fig. 3.2. Again we notice that all nonnegative solutions are positive in w0, 1x.

FIGURE 3.2

POSITIVE SOLUTIONS

101

Then building a quadrature method as was done in section 2, we get

uЈŽ x. 2

y 2

s FŽ uŽ x.. y FŽ . ;

x g Ž0, 1. Ž3.1.1.

and

uЈŽ x . s y'2 'F Ž . y F Ž uŽ x . . ; x g Ž0, 1. . Ž3.1.2.

Integrating Ž3.1.2. on Ž0, x., we have

HuŽ x.

ds

s y'2 x ;

' uŽ0. F Ž . y F Ž s.

x g Ž0, 1. .

Ž 3 .1 .3.

Let uЈŽ1. s ym where m ) 0. Then uŽ1. s mr␣ g Ž0, .. Substituting x s 1 in Ž3.1.3., we obtain

1

ds

H ' s

' '2 mr␣ F Ž . y F Ž s.

Ž 3 .1 .4.

and substituting x s 1 in Ž3.1.2., we get

'm

' s

.

'2 F Ž . y F Ž mr␣ .

Ž 3 .1 .5.

Combining Ž3.1.4. and Ž3.1.5., for such a solution to exist, there must exist an m such that

ds

m

Hs

.

' ' mr␣ F Ž . y F Ž s.

F Ž . y F Ž mr␣ .

Ž 3 .1 .6.

We now investigate whether such an m exists. Given g Ž , ϱ., if we define

ds

H GŽ m. s

' mr␣ F Ž . y F Ž s.

'm

and H Ž m. s

,

F Ž . y F Ž mr␣ .

then GŽ m. is a decreasing function of m. Further, HŽ0. s 0, HŽ m. ª ϱ as m ª ␣ and

2 ␣ F Ž . y F Ž mr␣ . q mf Ž mr␣ . HЈŽ m. s 2 ␣ F Ž . y F Ž mr␣ . 3r2 .

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ANURADHA ET AL.

Note that HЈŽ m. ) 0 for m G ␣. For m g Ž0, ␣ ., define kŽ m. s

2 ␣ w FŽ . y FŽ mr␣ .x q mfŽ mr␣ .. Then kŽ0. s 2 ␣ FŽ . ) 0, kЈŽ m. s

yfŽ mr␣ . q

m ␣

f ЈŽ mr␣

.

)

0

since

f Ž s. - 0 on Ž0,  . and

f ЈŽ s. G 0. Thus,

kŽ m. ) 0 on Ž0, ␣ . and hence HŽ m. is an increasing function of m. Thus

GŽ m. and HŽ m. has the shape as described in Fig. 3.3. Therefore there exists a unique mU0, 1 s mU0, 1Ž ␣ , . such that GŽ mU0, 1. s HŽ mU0, 1..

Now by back-tracking, one can prove the existence of positive solution

to Ž1.1.᎐Ž1.3. as described in Theorem 3.1.1.

THEOREM 3.1.1. Gi¨en ␣ g Ž0, ϱ. and g Ž , ϱ., there exists a unique mU0, 1 s mU0, 1Ž ␣ , . such that

ds

m

Hs

' ' mr␣ F Ž . y F Ž s.

F Ž . y F Ž mr␣ .

for m s mU0, 1. Also, there exists a unique s 0, 1Ž , mU0, 1. gi¨en by either

1

ds

H ' s

' '2 mr␣ F Ž . y F Ž s.

or

'm

' s '2 F Ž . y F Ž mr␣ .

for which Ž1.1.᎐Ž1.3. ha¨e a unique positi¨e solution uŽ x. as defined by

HuŽ x.

ds

s y'2 x ;

' uŽ0. F Ž . y F Ž s.

with uŽ0. s and uЈŽ1. s ymU0, 1.

x g Ž0, 1.

FIGURE 3.3

POSITIVE SOLUTIONS

103

COROLLARY 3.1.2. Gi¨en ␣ g Ž0, ϱ., the bifurcation diagram Ž , . of the positi¨e solutions of Ž1.1.᎐Ž1.3. with uŽ0. s ) is described by either

1

ds

H ' s

' '2 mr␣ F Ž . y F Ž s.

or

'm

' s '2 F Ž . y F Ž mr␣ .

for m s mU0, 1.

Since mU0, 1 g Ž0, ␣ ., we

' ' Ž . s 1 H0

ds

'2

FŽ . y FŽs.

solutions of the problem

have 0 - 0, 1Ž , mU0, 1. - Ž . where s describes the bifurcation diagram of positive

yuЉ Ž u. s f Ž uŽ x . . ; uЈŽ0. s 0 s uŽ1.

0 - x - 1 Žy1 - x - 1. Ž uŽy1. . s 0 suŽ1. . .

3.2. Existence Results when g w , .

We first note that there cannot be a nonnegative solution of Ž1.1.᎐Ž1.3. with uŽ0. s  for any . This follows from the fact that the unique solution of

yuЉ Ž x . s f Ž uŽ x . . ; 0 - x - 1

uЈŽ0. s 0, uŽ0. s 

is uŽ x. '  Žsince f Ž  . s 0. while uŽ x. '  does not satisfy Ž1.3.. Now assume that uŽ x. is a positive solution of Ž1.1.᎐Ž1.3. with uŽ0. s

g Ž , .. Since ) , u must be initially concave down. Also by Ž1.3. and the fact that s , we must have uЈŽ1. s y␣ uŽ1. - 0. ŽFor if uŽ1. s 0 then uЈŽ1. s 0 and hence by Ž3.2.1.. Žbelow. we get FŽ . s FŽ0. s 0, which is a contradiction to the fact that FŽ . / 0.. Thus u must have either none or an even number of critical points.

Now, in general we consider a solution with 2 n interior critical points where n s 0, 1, 2, . . . Žsee Fig. 3.4.. Note that since Ž1.1. is autonomous, every solution of Ž1.1.᎐Ž1.3. is symmetric about each of its critical points. Therefore it is enough to study solutions between w0, x0 x and w2 nx0, 1x, where x0 is the first interior critical point. Let uŽ0. s uŽ2 x0. s иии s uŽ2 nx0. s g Ž , .. Let r g Ž0,  . be the unique number such that FŽ . s FŽ r . Žsee Fig. 3.5.. Then uŽ x0. s uŽ3 x0. s иии s uŽŽ2 n y 1. x0.