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Positive Solutions for a Class of Nonlinear Boundary Value Problems with Neumann–Robin Boundary Conditions
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 V. Anuradha 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 yu⬙ Ž x . s f Ž u Ž x . . ;
0-x-1
u⬘ Ž 0 . s 0; u⬘ Ž 1 . q ␣ u Ž 1 . s 0 where ) 0 and ␣ ) 0 are parameters, and f g C 2 w0, 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. 䊚 1999 Academic Press
1. INTRODUCTION Consider the nonlinear boundary value problem yu⬙ Ž x . s f Ž u Ž x . . ;
0-x-1
Ž 1.1.
u⬘ Ž 0 . s 0;
Ž 1.2.
u⬘ Ž 1 . q ␣ u Ž 1 . s 0
Ž 1.3.
94 0022-247Xr99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
95
POSITIVE SOLUTIONS
where ) 0 and ␣ ) 0 are parameters and f g C 2 w0, 1x. For positone problems, since f Ž s . ) 0 for s G 0, for the case ␣ s 0 Ž u⬘Ž0. s 0 s u⬘Ž1.., 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 analyzing 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 f Ž s.
lim
s
sª⬁
s G 0, s ⬁,
f ⬘ Ž s . G 0, f ⬙ Ž s . ) 0, F Ž s. s
s
H0 f Ž t . dt.
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 u⬘Ž x ., we have yu⬙ Ž x . u⬘Ž x . s f Ž uŽ x .. u⬘Ž x . and integrating the above equation, we obtain y
u⬘ Ž x . 2
2
s F Ž u Ž x . . q C.
Since uŽ0. s , C s y F Ž uŽ0.. s y F Ž . and therefore y
u⬘ Ž x . 2
2
s F Ž uŽ x . . y F Ž . ;
x g Ž 0, 1 .
Ž 2.1.
96
ANURADHA ET AL.
FIGURE 2.1
and u⬘ Ž x . s y'2
'F Ž . y F Ž u Ž x . . ;
x g Ž 0, 1 . .
Ž 2.2.
x g Ž 0, 1 . .
Ž 2.3.
Integrating Ž2.2. on Ž0, x ., we have ds
HuuŽ0.x 'F Ž . y F Ž s . Ž .
s y'2 x ;
Let u⬘Ž1. s ym where m ) 0. Then uŽ1. s in Ž2.3., we obtain
'
s
1
m ␣
g Ž0, .. Substituting x s 1
ds
Ž 2.4.
'2 Hmr␣ 'F Ž . y F Ž s .
and substituting x s 1 in Ž2.2., we get
'
s
m
'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
Hmr␣ 'F Ž . y F Ž s .
s
m
'F Ž . y F Ž mr␣ .
We first investigate whether such an m exists.
.
Ž 2.6.
97
POSITIVE SOLUTIONS
For m g Ž0, ␣ ., define GŽ m. s Hm r ␣ H0
ds
'F Ž . y F Ž s .
ds
'F Ž . y F Ž s .
) 0, GŽ ␣ . s 0, and G⬘Ž m. s
. Then GŽ0. s
y1
␣ F Ž . y F Ž mr ␣ .
'
. Hence,
for a given ␣ g Ž0, ⬁. and g Ž0, ⬁., GŽ m. is a decreasing function of m m Žsee Fig. 2.2.. Let H Ž m. s ; then H Ž0. s 0, H Ž m. ª ⬁ as
'F Ž . y F Ž mr␣ .
m ª ␣ , and H⬘ Ž m . s
2 ␣ F Ž . y F Ž mr␣ . q mf Ž mr␣ . 2 ␣ F Ž . y F Ž mr␣ .
3r2
) 0.
Hence, for a given ␣ g Ž0, ⬁. and g Ž0, ⬁., H Ž m. is an increasing function of m Žsee Fig. 2.2.. Thus, given ␣ g Ž0, ⬁. and g Ž0, ⬁., clearly there exists a unique m s m*Ž ␣ , . g Ž0, ␣ . such that GŽ m*. s H Ž 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, ⬁. and g Ž0, ⬁., there exists a unique m s m*Ž ␣ , . g Ž0, ␣ . such that ds
Hmr␣ 'F Ž . y F Ž s .
s
m
'F Ž . y F Ž mr␣ .
for m s m*. Also, there exists a unique s Ž , m*. gi¨ en by either
'
s
1
'2
ds
Hmr␣ 'F Ž . y F Ž s .
FIGURE 2.2
98
ANURADHA ET AL.
or
'
s
m
'2 'F Ž . y F Ž mr␣ .
for which Ž1.1. ᎐ Ž1.3. has a unique positi¨ e solution uŽ x . as defined by ds
HuuŽ0.x 'F Ž . y F Ž s . Ž .
s y'2 x ;
x g Ž 0, 1 .
with uŽ0. s and u⬘Ž1. s ym*. COROLLARY 2.2. Gi¨ en ␣ g Ž0, ⬁., the bifurcation diagram Ž , . of the positi¨ e solutions of Ž1.1. ᎐ Ž1.3. is described by either
'
s
1
'2
ds
Hmr␣ 'F Ž . y F Ž s .
or
'
s
m
'2 'F Ž . y F Ž mr␣ .
for m s m*. Since m* g Ž0, ␣ ., we have 0 - Ž , m*. - Ž . where 1
'2
H0
ds
'F Ž . y F Ž s .
' Ž .
s
. Note that s Ž . describes the bifurcation diagram
of positive solutions of the boundary value problem yu⬙ Ž x . s f Ž u Ž x . . ; u⬘ Ž 0 . s 0 s u Ž 1 .
0-x-1
Ž y1 - x - 1 .
Ž u Ž y1. . s 0 su Ž 1 . . .
f Ž s.
Since lim s ª⬁ s ⬁ it follows that lim ª⬁ Ž . s 0. Hence, given s Ž ) 0 and ␣ g 0, ⬁., 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, ⬁., Ž1.1. ᎐ Ž1.3. ha¨ e at least one positi¨ e solution with no interior critical points.
99
POSITIVE SOLUTIONS
FIGURE 2.3
3. EXISTENCE RESULTS WHEN f Ž0. - 0 Throughout this section we will assume the following: ᭚  , ) 0 such f s
Ž . that f Ž s . - 0 on w0,  ., f Ž  . s 0, f ⬘Ž s . G 0, f ⬙ Ž s . ) 0, lim s ª⬁ s ⬁, s s and F Ž . s 0 where F Ž s . s H0 f Ž t . 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 2 Ž , n. for which Ž1.1. ᎐ Ž1.3. have nonnegative solutions with n interior critical points. We denote these solutions by u n, 1Ž , 1 , ␣ . and u n, 2 Ž , 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 0, 4 then necessarily u⬘Ž1. - 0, while if g 0, 4 then for each n s 0, 1, 2, . . . , there exists a solution with u⬘Ž1. - 0 at s 1 and a second solution with u⬘Ž1. 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 u 0, 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 describing our results. 3.1. Existence Results when ) Let uŽ x . be a nonnegative solution of Ž1.1. ᎐ Ž1.3. with uŽ0. s g Ž , ⬁.. Since )  and f ) 0 for u )  , u must be initially concave down. Further, there does not exist 0 F r - such that F Ž . s F Ž 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
101
POSITIVE SOLUTIONS
Then building a quadrature method as was done in section 2, we get u⬘ Ž x .
y
2
s F Ž uŽ x . . y F Ž . ;
2
x g Ž 0, 1 .
Ž 3.1.1.
x g Ž 0, 1 . .
Ž 3.1.2.
and u⬘ Ž x . s y'2
'F Ž . y F Ž u Ž x . . ;
Integrating Ž3.1.2. on Ž0, x ., we have ds
HuuŽ0.x 'F Ž . y F Ž s . Ž .
s y'2 x ;
x g Ž 0, 1 . .
Ž 3.1.3.
Let u⬘Ž1. s ym where m ) 0. Then uŽ1. s mr␣ g Ž0, .. Substituting x s 1 in Ž3.1.3., we obtain
'
s
1
ds
Ž 3.1.4.
'2 Hmr␣ 'F Ž . y F Ž s .
and substituting x s 1 in Ž3.1.2., we get
'
s
m
'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
Hmr␣ 'F Ž . y F Ž s .
s
m
'F Ž . y F Ž mr␣ .
.
Ž 3.1.6.
We now investigate whether such an m exists. Given g Ž , ⬁., if we define GŽ m. s
ds
Hmr␣ 'F Ž . y F Ž s .
and
H Ž m. s
m
'F Ž . y F Ž mr␣ .
,
then GŽ m. is a decreasing function of m. Further, H Ž0. s 0, H Ž m. ª ⬁ as m ª ␣ and H⬘ Ž m . s
2 ␣ F Ž . y F Ž mr␣ . q mf Ž mr␣ . 2 ␣ F Ž . y F Ž mr␣ .
3r2
.
102
ANURADHA ET AL.
Note that H⬘Ž m. ) 0 for m G ␣ . For m g Ž0, ␣ ., define k Ž m. s 2 ␣ w F Ž . y F Ž mr␣ .x q mf Ž mr␣ .. Then k Ž0. s 2 ␣ F Ž . ) 0, k⬘Ž m. s yf Ž mr␣ . q m␣ f ⬘Ž mr␣ . ) 0 since f Ž s . - 0 on Ž0,  . and f ⬘Ž s . G 0. Thus, k Ž m. ) 0 on Ž0, ␣ . and hence H Ž m. is an increasing function of m. Thus GŽ m. and H Ž 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 H Ž 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, ⬁. and g Ž , ⬁., there exists a unique s mU0, 1Ž ␣ , . such that
mU0, 1
ds
Hmr␣ 'F Ž . y F Ž s .
s
m
'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
'
s
1
ds
'2 Hmr␣ 'F Ž . y F Ž s .
or
'
s
m
'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 ds
HuuŽ0.x 'F Ž . y F Ž s . Ž .
s y'2 x ;
with uŽ0. s and u⬘Ž1. s ymU0, 1.
FIGURE 3.3
x g Ž 0, 1 .
103
POSITIVE SOLUTIONS
COROLLARY 3.1.2. Gi¨ en ␣ g Ž0, ⬁., the bifurcation diagram Ž , . of the positi¨ e solutions of Ž1.1. ᎐ Ž1.3. with uŽ0. s ) is described by either
'
s
1
ds
'2 Hmr␣ 'F Ž . y F Ž s .
or
'
s
m
'2 'F Ž . y F Ž mr␣ .
for m s mU0, 1. Since mU0, 1 g Ž0, ␣ ., we have 0 - 0, 1Ž , mU0, 1 . - Ž . where s
' Ž . s '12 H
0
ds
'F Ž . y F Ž s .
describes the bifurcation diagram of positive
solutions of the problem yu⬙ Ž u . s f Ž u Ž x . . ; u⬘ Ž 0 . s 0 s u Ž 1 .
0-x-1
Ž y1 - x - 1 .
Ž u Ž y1. . s 0 su Ž 1 . . .
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 yu⬙ Ž x . s f Ž u Ž x . . ; u⬘ Ž 0 . s 0,
0-x-1
u Ž 0. 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 u⬘Ž1. s y␣ uŽ1. - 0. ŽFor if uŽ1. s 0 then u⬘Ž1. s 0 and hence by Ž3.2.1.. Žbelow. we get F Ž . s F Ž0. s 0, which is a contradiction to the fact that F Ž . / 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, x 0 x and w2 nx 0 , 1x, where x 0 is the first interior critical point. Let uŽ0. s uŽ2 x 0 . s ⭈⭈⭈ s uŽ2 nx 0 . s g Ž  , .. Let r g Ž0,  . be the unique number such that F Ž . s F Ž r . Žsee Fig. 3.5.. Then uŽ x 0 . s uŽ3 x 0 . s ⭈⭈⭈ s uŽŽ2 n y 1. x 0 .
104
ANURADHA ET AL.
FIGURE 3.4
FIGURE 3.5
s r. Thus we see that a nonnegative solution of Ž1.1. ᎐ Ž1.3. with uŽ0. s g Ž  , . and 2 n interior critical points, where n s 0, 1, 2, . . . , is positive in w0, 1x. Building a quadrature method as before we see that
y
u⬘ Ž x . 2
2
s F Ž uŽ x . . y F Ž . ;
x g Ž 0, 1 .
Ž 3.2.1.
x g Ž 0, x 0 .
Ž 3.2.2.
which gives u⬘ Ž x . s y'2
'F Ž . y F Ž u Ž x . . ;
105
POSITIVE SOLUTIONS
and u⬘ Ž x . s y'2
'F Ž . y F Ž u Ž x . . ;
x g Ž 2 nx 0 , 1 . . Ž 3.2.3.
Integrating Ž3.2.2. and Ž3.2.3. on Ž0, x. and Ž2 nx 0 , 1. respectively, we get ds
HuuŽ0.x 'F Ž . y F Ž s . Ž .
s y'2 x ;
x g Ž 0, x 0 .
Ž 3.2.4.
and ds
HuuŽ x1. 'F Ž . y F Ž s . Ž .
s y'2 Ž 1 y x . ;
x g Ž 2 nx 0 , 1 . . Ž 3.2.5.
Let u⬘Ž1. s ym where m ) 0. Then uŽ1. s m␣ g Ž r, .. Substituting x s x 0 in Ž3.2.4. and x s 2 nx 0 in Ž3.2.5. and solving the resulting equations we get x0 s
ž 'F Ž . y F Ž s . / dsr ž 'F Ž . y F Ž s . / q H dsr ž 'F Ž . y F Ž s . / Hr dsr
2 nHr
r
m r␣
Ž 3.2.6. and
'
s
2n
ds
'2 Hr 'F Ž . y F Ž s .
q
1
ds
'2 Hmr␣ 'F Ž . y F Ž s .
. Ž 3.2.7.
Further, substituting x s in Ž3.2.3., we get
'
s
m
'2 'F Ž . y F Ž mr␣ .
.
Ž 3.2.8.
Combining Ž3.2.7. and Ž3.2.8., for such a solution to exist, there must exist an m such that 2n
ds
Hr 'F Ž . y F Ž s .
q
ds
Hmr␣ 'F Ž . y F Ž s .
s
m
'F Ž . y F Ž mr␣ .
.
Ž 3.2.9. We now investigate, as before, if such an m exists.
106
ANURADHA ET AL.
For m g Ž ␣ r, ␣ ., define GŽ m. s
ds
Hmr␣ 'F Ž . y F Ž s . H Ž m. s
ds
q 2n
and
Hr 'F Ž . y F Ž s .
m
'F Ž . y F Ž mr␣ .
.
Then GŽ m. is a decreasing function of m. Now H Ž m. ª ⬁ as m ª ␣ r, ␣ and H⬘ Ž m . s
2 ␣ F Ž . y F Ž mr␣ . q mf Ž mr␣ . 2 ␣ F Ž . y F Ž mr␣ .
3r2
.
Then H⬘Ž m. ) 0 for m G ␣ . For m g Ž ␣ r, ␣ . define hŽ m. s 2 ␣ w F Ž . y F Ž mr␣ .x q mf Ž mr␣ .. Then hŽ ␣ r . s ␣ rf Ž r . - 0 Žsince r -  ., hŽ ␣ . s 2 ␣ w F Ž . y F Ž  .x ) 0 and h⬘Ž m. s yf Ž mr␣ . q m Ž . Ž . Ž . Ž . ␣ f ⬘ mr␣ ) 0 since f s - 0 on r,  and f ⬘ s G 0. Thus there exists a unique c g Ž ␣ r, ␣ . such that hŽ c . s 0. Thus H Ž m. and GŽ m. have one of the forms described in Fig. 3.6. Now, to investigate whether H Ž m. and GŽ m. intersect, we define LŽ m. s H Ž m. y GŽ m. Žsee Fig. 3.7.. That is, LŽ m. s
m
'F Ž . y F Ž mr␣ . y 2n
ds
y
Hr 'F Ž . y F Ž s .
ds
Hmr␣ 'F Ž . y F Ž s . .
Then LŽ m. ª ⬁ as m ª ␣ r, ␣ and L⬘ Ž m . s
2 Ž ␣ q 1 . F Ž . y F Ž mr␣ . q mf Ž mr␣ . 2 ␣ F Ž . y F Ž mr␣ .
3r2
.
It is easy to see that L⬘Ž m. ) 0 on w ␣ , ␣ .. For m g Ž ␣ r, ␣ . define k Ž m. s 2Ž ␣ q 1.w F Ž . y F Ž mr␣ .x q mf Ž mr␣ .. Then k Ž ␣ r . s ␣ rf Ž r . 0 Žsince r -  ., k Ž ␣ . s 2Ž ␣ q 1.w F Ž . y F Ž  .x ) 0, and k⬘Ž m. s ␣q 2 Žyf Ž mr␣ .. q Ž mr␣ . f ⬘Ž mr␣ . ) 0. Thus there exists a unique c g ␣ Ž ␣ r, ␣ . such that k Ž c . s 0, i.e., L⬘Ž c . s 0. Note that Ž1.1. ᎐ Ž1.3. have no positive solution if LŽ c . ) 0. Since k Ž c . s 0, we have F Ž . y F Ž ␣c . s
POSITIVE SOLUTIONS
FIGURE 3.6
FIGURE 3.7
107
108 y
ANURADHA ET AL.
c 2Ž ␣ q 1.
f Ž ␣c .. So LŽ c . s
c
'F Ž . y F Ž cr␣ . y 2n
s
y
(
ds
Hcr␣ 'F Ž . y F Ž s .
ds
Hr 'F Ž . y F Ž s .
2 c Ž ␣ q 1. yf Ž cr␣ .
y
ds
y 2n
ds
Hcr␣ 'F Ž . y F Ž s .
Hr 'F Ž . y F Ž s .
.
Since c g Ž ␣ r, ␣ ., we have
(
2 c Ž ␣ q 1. yf Ž cr␣ .
y Ž 2 n q 1.
ds
Hr 'F Ž . y F Ž s .
F LŽ c . F
(
2 c Ž ␣ q 1. yf Ž cr␣ .
y
ds
y 2n
H 'F Ž . y F Ž s .
ds
Hr 'F Ž . y F Ž s .
.
Ž 3.2.10. Using the right inequality of Ž3.2.10. we see that LŽ c . F
F
F
( ( (
2 c Ž ␣ q 1. yf Ž cr␣ . 2 c Ž ␣ q 1. yf Ž cr␣ . 2 c Ž ␣ q 1. yf Ž cr␣ .
ds
y Ž 2 n q 1.
H 'F Ž . y F Ž s .
y Ž 2 n q 1.
H 'F Ž . y F Ž  .
y
ds
Ž 2 n q 1. Ž y  . 'F Ž . y F Ž  . .
We see that as ª , r ª 0, and k Ž ␣ r . s ␣ rf Ž r . ª 0y and so c ª 0. Thus lim ª LŽ c . G y
Ž2 n q 1 . Ž y  .
'y F Ž  .
- 0. Similarly, using the left in-
109
POSITIVE SOLUTIONS
equality of Ž3.2.10. we get
LŽ c . G
(
2 c Ž ␣ q 1. yf Ž cr␣ .
y Ž 2 n q 1.
ds
Hr 'F Ž . y F Ž s .
.
But as ª  , r ª  and c ª ␣ . Thus lim ª  LŽ c . s ⬁. Hence we see that LŽ c . - 0 for close to and LŽ c . ) 0 for close to  . Therefore, there exists an a*n g Ž  , . such that for each g Ž aUn , ., LŽ cŽ .. - 0. Therefore, for each g Ž aUn , ., there exists mU2 n, 1 g Ž cŽ ., ␣ . and mU2 n, 2 g Ž ␣ r, cŽ .. such that LŽ mU2 n, 1 . s 0 s LŽ mU2 n, 2 .. For s aUn we have LŽ cŽ aUn .. s 0 and so there exists a unique mU2 n, 1 g Ž ␣ r, ␣ . such that LŽ mU2 n, 1 . s 0. Further there exists bnU g Ž  , aUn x such that for g w  , bnU . there are no m’s such that LŽ m. s 0. Note also that aUn ) aUnq1 for each n s 0, 1, 2, . . . . Now, by back-tracking one can prove the existence of such solutions of Ž1.1. ᎐ Ž1.3. as described in the Theorem 3.2.1 below. THEOREM 3.2.1. Let n s 0, 1, 2, . . . , ␣ g Ž0, ⬁. and let mU2 n, 1 , mU2 n, 2 , bnU be as described abo¨ e.
aUn ,
ŽA. Then for g Ž aUn , . Ž1.1. ᎐ Ž1.3. ha¨ e exactly two solutions u 2 n, i , i s 1, 2 with 2 n interior critical points at s 2 n, i Ž , mU2 n, i ., i s 1, 2 respecti¨ ely. Here 2 n, i Ž , mU2 n, 1 ., i s 1, 2 are gi¨ en by
'
s
2n
ds
'2 Hr 'F Ž . y F Ž s .
q
1
ds
'2 Hmr␣ 'F Ž . y F Ž s .
or
'
s
m
'2 'F Ž . y F Ž mr␣ .
.
The corresponding solutions are defined by ds
HuuŽ0.x 'F Ž . y F Ž s . Ž .
ds
HuuŽ x1. 'F Ž . y F Ž s . Ž .
s y'2 x ;
s y'2 Ž 1 y x . ;
x g Ž 0, x 0 . , x g Ž 2 nx 0 , 1 .
110
ANURADHA ET AL.
where x 0 s x 0 Ž , mU2 n, i . is gi¨ en by x0 s
ž 'F Ž . y F Ž s . / dsr ž 'F Ž . y F Ž s . / q H dsr ž 'F Ž . y F Ž s . / Hr dsr
2 nHr
r
m r␣
.
Further, 2 n, 1Ž , mU2 n, 1 . - 2 n, 2 Ž , mU2 n, 2 .. ŽB. For s aUn , Ž1.1. ᎐ Ž1.3. ha¨ e exactly one solution u 2 n, 1 with 2 n interior critical points at s 2 n, 1Ž aUn , mU2 n, 1 . where 2 n, 1 and u 2 n, 1 are as defined in ŽA.. ŽC. For g w  , bnU ., Ž1.1. ᎐ Ž1.3. ha¨ e no solution with 2 n interior critical points for any . COROLLARY 3.2.2. Gi¨ en ␣ g Ž0, ⬁., the bifurcation diagram Ž , . of positi¨ e solutions with 2 n interior critical point of Ž1.1. ᎐ Ž1.3. with uŽ0. s g w  , . is described by
'
s
2n
'2
ds
Hr 'F Ž . y F Ž s .
q
1
'2
ds
Hmr␣ 'F Ž . y F Ž s .
or
'
s
m
'2 'F Ž . y F Ž mr␣ .
for m s mU2 n, 1 and m s mU2 n, 2 . 3.3. Existence Results when g Ž0,  . First we note that if u is a nonnegative solution of Ž1.1. ᎐ Ž1.3. with Ž u 0. s g Ž0,  ., u must have at least one critical point Žsince u⬘Ž1. - 0.. In fact, u must have an odd number of interior critical points. In general, we consider a nonnegative solution with 2 n q 1, n s 0, 1, 2, . . . interior critical points Žsee Fig. 3.8.. Again, since Ž1.1. is autonomous, it is enough to study a solution in w0, x 1 x and wŽ2 n q 1. x 1 , 1x where x 1 is the first interior critical point of u. Suppose uŽ0. s uŽ2 x 1 . s ⭈⭈⭈ s uŽ2 nx 1 . s g Ž0,  .. Let r g Ž  , . be such that F Ž . s F Ž r .. Then uŽ x 1 . s uŽ3 x 1 . s ⭈⭈⭈ s uŽŽ2 n q 1. x 1 . s r. In this case also, all nonnegative solutions are positive in w0, 1x. Building a quadrature method as before, we see that y
u⬘ Ž x . 2
2
s F Ž uŽ x . . y F Ž . ;
x g Ž 0, 1 .
Ž 3.3.1.
111
POSITIVE SOLUTIONS
FIGURE 3.8
which gives u⬘ Ž x . s '2
'F Ž . y F Ž u Ž x . . ;
x g Ž 0, x 1 .
Ž 3.3.2.
and u⬘ Ž x . s y'2
'F Ž . y F Ž u Ž x . . ;
x g Ž Ž 2 n q 1 . x 1 , 1 . . Ž 3.3.3.
Integrating Ž3.3.2. and Ž3.3.3. on Ž0, x 1 . and ŽŽ2 n q 1. x 1 , 1., respectively, we get ds
HuuŽ0.x 'F Ž . y F Ž s . Ž .
s '2 x ;
x g Ž 0, x 1 .
Ž 3.3.4.
and ds
HuuŽ x1. 'F Ž . y F Ž s . Ž .
s y'2 Ž 1 y x . ;
x g Ž Ž 2 n q 1. x1 , 1. .
Ž 3.3.5. Let u⬘Ž1. s ym where m ) 0. Then uŽ1. s m␣ g Ž , r .. Substituting x s x 1 in Ž3.3.4. and x s Ž2 n q 1. x 1 in Ž3.3.5. and solving the resulting equations we get x1 s
ž 'F Ž . y F Ž x . / dsr ž 'F Ž . y F Ž s . / y H dsr ž 'F Ž . y F Ž s . / Hr dsr
Ž 2 n q 1 . Hr
r
m r␣
Ž 3.3.6.
112
ANURADHA ET AL.
and
'
s
Ž 2 n q 1.
'2
ds
r
q
H 'F Ž . y F Ž s .
1
ds
r
'2
Hmr␣ 'F Ž . y F Ž s .
.
Ž 3.3.7. Further, substituting x s 1 in Ž3.3.3., we get
'
m
s
'F Ž . y F Ž mr␣ .
.
Ž 3.3.8.
Combining Ž3.3.7. and Ž3.3.8., for such a solution to exist, there must exist an m such that
Ž 2 n q 1. H
s
ds
r
'F Ž . y F Ž s . m
'F Ž . y F Ž mr␣ .
q
ds
Hmr␣ 'F Ž . y F Ž s .
.
Ž 3.3.9.
We now investigate, as before, if such an m exists. For m g Ž ␣ , ␣ r ., define GŽ m. s
r
ds
Hmr␣ 'F Ž . y F Ž s . H Ž m. s
q Ž 2 n q 1.
ds
r
H 'F Ž . y F Ž s .
m
'F Ž . y F Ž mr␣ .
and
.
Then following the similar argument as in Section 3.2 one can see that GŽ m. and H Ž m. have the form as in Fig. 3.4 Žwith r and interchanged.. To investigate whether GŽ m. and H Ž m. intersect, we define P Ž m. s H Ž m. y GŽ m.. That is, P Ž m. s
m
'F Ž . y F Ž mr␣ . y Ž 2 n q 1.
r
y ds
r
ds
Hmr␣ 'F Ž . y F Ž s .
H 'F Ž . y F Ž s .
113
POSITIVE SOLUTIONS
for m g Ž ␣ , ␣ r .. Then P Ž m. ª ⬁ as m ª ␣ , ␣ r and P⬘ Ž m . s
2 Ž ␣ q 1 . F Ž . y F Ž mr␣ . q mf Ž mr␣ . 3r2
2 ␣ F Ž . y F Ž mr␣ .
s L⬘ Ž m . .
Using a similar argument as in Section 3.2, one can see that there exists a unique c g Ž ␣ , ␣ . such that P⬘Ž c . s 0 s L⬘Ž c .. Now r c ds P Ž c. s q F Ž . y F Ž cr␣ . cr ␣ F Ž . y F Ž s .
H '
'
y Ž 2 n q 1.
ds
r
H 'F Ž . y F Ž s .
.
Since c g Ž ␣ , ␣ ., we have
(
2 c Ž ␣ q 1. yf Ž cr␣ .
y Ž 2 n q 1.
F P Ž c. F
(
2 c Ž ␣ q 1. yf Ž cr␣ .
y Ž 2 n q 1.
ds
r
H 'F Ž . y F Ž s .
r
y
ds
r
H 'F Ž . y F Ž s . ds
H 'F Ž . y F Ž s .
.
As before, we can show that lim ª 0 P Ž m. s lim r ª P Ž m. - 0 and lim ª  P Ž m. s lim r ª  P Ž m. s ⬁. Therefore there exists an a ˜n g Ž0,  . Ž . Ž Ž .. such that for each g 0, a ˜n , P c - 0. Thus for each g Ž0, a˜n ., U U there exists m 2 nq1, 1 s m 2 nq1, 1Ž ␣ , . g Ž cŽ ., ␣ r . and mU2 ny1, 2 s mU2 nq1, 1Ž ␣ , . g Ž ␣ , cŽ .. such that P Ž mU2 nq1, 1 . s 0 s P Ž mU2 nq1, 2 .. For s a˜n , we have LŽ cŽ a˜n .. s 0 and so there exists a unique mU2 nq1, 1 g Ž ␣ , ␣ r . such that P Ž mU2 nq1, 1 . s 0. Further, there exists ˜ bn g w a ˜n ,  . ˜ Ž . Ž . such that for g bn ,  there are no m’s such that P m s 0. Note again that a ˜n - a˜nq1 for each n s 0, 1, 2, . . . . Now, by back-tracking one can prove the existence of such solutions of Ž1.1. ᎐ Ž1.3. as described in Theorem 3.3.1 below. THEOREM 3.3.1. Let n s 0, 1, 2, . . . , ␣ g Ž0, ⬁. and let mU2 nq1, 1 , mU2 nq1, 2 , a ˜n , ˜bn be as described abo¨ e. ŽA. Then for g Ž0, a ˜n . Ž1.1. ᎐ Ž1.3. ha¨ e exactly two solutions u 2 nq1, i , i s 1, 2 with 2 n q 1 interior critical points at s 2 nq1, i Ž , mU2 nq1, i ., i s 1, 2 respecti¨ ely. Here 2 n, i Ž , mU2 n, i ., i s 1, 2 are gi¨ en by 2n q 1 r ds 1 r ds ' s q '2 F Ž . y F Ž s . '2 mr␣ F Ž . y F Ž s .
H'
H '
114
ANURADHA ET AL.
or
'
s
m
'2 'F Ž . y F Ž mr␣ .
.
The corresponding solutions are defined by ds
HuuŽ0.x 'F Ž . y F Ž s . Ž .
s '2 x ;
x g Ž 0, x 1 .
and ds
HuuŽ x1. 'F Ž . y F Ž s . Ž .
s y'2 Ž 1 y x . ;
x g Ž Ž 2 n q 1. x1 , 1.
where x 1 s x 1Ž , mU2 nq1, i . is gi¨ en by x1 s
ž 'F Ž . yF Ž s . / dsr ž 'F Ž . yF Ž s . / yH dsr ž 'F Ž . yF Ž s . / Hr dsr
Ž 2 nq1 . Hr
r
m r␣
.
Further, 2 nq1, 1Ž , mU2 nq1, 1 . - 2 nq1, 2 Ž , mU2 nq1, 2 .. ŽB. For s a ˜n , Ž1.1. ᎐ Ž1.3. ha¨ e exactly one solution u 2 nq1, 1 with 2 n q 1 interior critical points at s 2 nq1, 1Ž a ˜n , mU2 nq1, 1 . where 2 nq1, 1 and u 2 nq1, 1 are as defined in ŽA.. ŽC. For g Ž ˜ bn ,  ., Ž1.1. ᎐ Ž1.3. ha¨ e no solution with 2 n q 1 interior critical points for any . COROLLARY 3.2.2. Gi¨ en ␣ g Ž0, ⬁., the bifurcation diagram Ž , . of positi¨ e solutions with 2 n q 1 interior critical points of Ž1.1. ᎐ Ž1.3. with uŽ0. s p⑀ Ž0,  . is described by
'
s
2n q 1
'2
ds
r
H 'F Ž . y F Ž s .
q
1
r
or
'
s
m
'F Ž . y F Ž mr␣ .
for m s mU2 nq1, 1 and m s mU2 nq1, 2 .
ds
'2 Hmr␣ 'F Ž . y F Ž s .
115
POSITIVE SOLUTIONS
3.4. Existence Results when s and s 0 First we will study the case s . Let u be a nonnegatiive solution of Ž1.1. ᎐ Ž1.3. with uŽ0. s . Since uŽ0. s )  , u is initially concave down. Note that if uŽ1. s 0 then u⬘Ž1. s 0 and if uŽ1. / 0 and then u⬘Ž1. s ym - 0. Hence u can have either none or an even number of interior critical points. We will first discuss the case when u⬘Ž1. s ym - 0 with 2 n interior critical points, where n s 0, 1, 2, . . . ŽFig. 3.9a.. Since Ž1.1. is autonomous, as in Sections 3.2 and 3.3, it is enough to study the solution in w0, x 0 x and w2 nx 0 , 1x where x 0 is the first interior critical point of u. Let uŽ0. s uŽ2 x 0 . s ⭈⭈⭈ s uŽ2 nx 0 . s . Then uŽ x 0 . s uŽ3 x 0 . s ⭈⭈⭈ s uŽŽ2 n y 1. x 0 . s 0. Again building a quadrature method, we see that
y
u⬘ Ž x . 2
2
s F Ž uŽ x . . ;
FIGURE 3.9
x g Ž 0, 1 .
Ž 3.4.1.
116
ANURADHA ET AL.
which gives u⬘ Ž x . s y 2 F Ž u Ž x . . ;
'
x g Ž 0, x 0 .
Ž 3.4.2.
u⬘ Ž x . s y 2 F Ž u Ž x . . ;
x g Ž 2 nx 0 , 1 . .
Ž 3.4.3.
and
'
Integrating Ž3.4.2. and Ž3.4.3. on Ž0, x 0 . and Ž2 nx 0 , 1., respectively, we get ds
HuuŽ0.x 'y F Ž s . Ž .
s y'2 x ;
x g Ž 0, x 0 .
Ž 3.4.4.
and ds
HuuŽ x1. 'y F Ž s . Ž .
s y'2 Ž 1 y x . ;
x g Ž 2 nx 0 , 1 . .
Ž 3.4.5.
Let u⬘Ž1. s ym where m ) 0. Then uŽ1. s m␣ g Ž0, .. Substituting x s x 0 in Ž3.4.4. and x s 2 nx 0 in Ž3.4.5. and solving the resulting equations we get H0 dsr
x0 s
2 nH0
dsr
ž 'y F Ž s . /
ž 'y F Ž s . / q Hm r ␣ dsr
Ž 3.4.6.
ž 'y F Ž s . /
and
'
s
2n
'2
ds
H0 'y F Ž s .
q
1
'2
ds
Hmr␣ 'y F Ž s .
.
Ž 3.4.7.
Further, substituting x s 1 in Ž3.4.3., we get
'
s
m
'2 'y F Ž mr␣ .
.
Ž 3.4.8.
Combining Ž3.4.7. and Ž3.4.8., for such a solution to exist, there must exist an m such that 2n
ds
H0 'y F Ž s .
q
ds
Hmr␣ 'y F Ž s .
s
m
'y F Ž mr␣ .
We now investigate, as before, if such an m exists.
.
Ž 3.4.9.
117
POSITIVE SOLUTIONS
For m g Ž0, ␣ ., define GŽ m. s
ds
Hmr␣ 'y F Ž s . H Ž m. s
q 2n
ds
and
H0 'y F Ž s .
m
'2 'y F Ž mr␣ .
.
Then GŽ m. is a decreasing function of m. Now H Ž0. s 0, H Ž m. ª ⬁ as m ª ␣ and H⬘ Ž m . s
2 ␣ yF Ž mr␣ . q mf Ž mr␣ . 2 ␣ yF Ž mr␣ .
3r2
.
Then H⬘Ž m. ) 0 for m G ␣ . For m g Ž0, ␣ . define hŽ m. s y2 ␣ w F Ž mr␣ .x q mf Ž mr␣ .. Then hŽ0. s 0 and hŽ ␣ . s y2 ␣ F Ž  . ) 0 and h⬘Ž m. s yf Ž mr␣ . q m␣ ) 0 since f Ž s . - 0 on Ž0,  . and f ⬘Ž2. G 0. Thus H Ž m. is an increasing function of m Žsee Fig. Ž3.10. and hence there exists a unique mU2 n, 1 s mU2 n, 1Ž ␣ , . g Ž0, ␣ . such that GŽ mU2 n, 1 . s H Ž mU2 n, 1 .. Now, by back-tracking one can prove the existence of nonnegative solutions with 2 n, n s 0, 1, 2, . . . interior critical points of Ž1.1. ᎐ Ž1.3. defined by Ž3.4.4. ᎐ Ž3.4.6. with uŽ0. s , u⬘Ž1. s ymU2 n, 1 at s 2 n, 1Ž , mU2 n, 1 . given by Ž3.4.7. Žor Ž3.4.8... We next discuss a solution satisfying uŽ0. s and uŽ1. s u⬘Ž1. s 0 with 2 n interior critical points, where n s 0, 1, 2, . . . Žsee Fig. 3.9b.. Here x 0 s 1rŽ2 n q 1. and the problem is equivalent to studying a positive
FIGURE 3.10
118
ANURADHA ET AL.
solution to yu⬙ Ž x . s f Ž u Ž x . . ;
x g Ž 0, x 0 .
Ž or Ž yx 0 , x 0 . . Ž or u Ž yx 0 . s 0 s u Ž x 0 . . .
u⬘ Ž 0 . s 0 s u Ž x 0 .
Here the quadrature method yields explicit relationships and it follows easily that such a solution defined by ds
H0u x 'y F Ž s . Ž .
s y'2 x ;
x g Ž 0, x 0 .
Ž 3.4.10.
exists at s 2 n, 2 Ž , 0., where
'
2 n, 2
Ž , 0. s
2n q 1
'2
ds
H0 'y F Ž s .
.
Ž 3.4.11.
Now we summarize the above information on solutions with 2 n, n s 0, 1, 2, . . . interior critical points in the following theorem. THEOREM 3.4.1 Ža.. Let n s 0, 1, 2, . . . , ␣ g Ž0, ⬁. and let mU2 n, 1 be as described abo¨ e. For s , Ž1.1. ᎐ Ž1.3. ha¨ e exactly two nonnegati¨ e solutions u 2 n, i , i s 1, 2 with 2 n interior critical points at s 2 n, 1Ž , mU2 n, 1 . and s 2 n, 2 Ž , 0., respecti¨ ely. Here 2 n, 1Ž , mU2 n, 1 . is gi¨ en by
'
s
2n
ds
'2 H0 'y F Ž s .
q
1
ds
'2 Hmr␣ 'y F Ž s .
or
'
s
m
'2 'y F Ž mr␣ .
.
The corresponding solution is gi¨ en by ds
HuuŽ0.x 'y F Ž s . Ž .
u Ž1 .
H
uŽ x .
ds
'y F Ž s .
s y'2 x ;
s y'2 Ž 1 y x . ;
x g Ž 0, x 0 . ,
x g Ž 2 nx 0 , 1 .
119
POSITIVE SOLUTIONS
where x 0 s x 0 Ž , mU2 n, 1 . is gi¨ en by x0 s
H0 dsr 2 nH0
dsr
ž 'y F Ž s . /
ž 'y F Ž s . / q Hm r ␣ dsr
ž 'y F Ž s . /
.
Also 2 n, 2 Ž , 0. is gi¨ en by
'
2 n, 2
Ž , 0. s
2n q 1
'2
0
ds
H 'y F Ž s .
and the corresponding solution is defined by ds
H0u x 'y F Ž s . Ž .
s y'2 x ;
x g Ž 0, x 0 .
where x 0 s 1rŽ2 n q 1.. Further note that 2 n, 1Ž , mU2 n, 1 . - 2 n, 2 Ž , 0. 2Ž nq1., 1Ž ,mU2Ž nq1., 1 .. Now suppose u is a nonnegative solution of Ž1.1. ᎐ Ž1.3. with uŽ0. s 0. Note that if uŽ1. s 0, u⬘Ž1. s 0 and if u⬘Ž1. / 0 the u⬘Ž1. - 0. Thus u must have an odd number of interior critical points in either case. Hence the solutions are of forms as described in Fig. 3.11. We will first discuss the case when u⬘Ž1. s ym - 0 with 2 n q 1 interior critical points, where n s 0, 1, 2, . . . Žsee Fig. 3.11a.. Since u⬘Ž1. s ym - 0, uŽ1. s mr␣ . Note that in order to study a nonnegative solution with 2 n q 1 interior critical points, due to the symmetry it is enough to study the solution in the intervals w0, x 1 x and wŽ2 n q 1. x 1 , 1x, where x 1 is the first interior critical point of u. Suppose uŽ0. s uŽ2 x 1 . s uŽ4 x 1 . s ⭈⭈⭈ s uŽ2 nx 1 . s 0. Then uŽ x 1 . s uŽ3 x 1 . s ⭈⭈⭈ s uŽŽ2 n q 1. x 1 . s . Building a quadrature method as before, we get y
u⬘ Ž x .
2
2
s F Ž uŽ x . . ;
x g Ž 0, 1 .
Ž 3.4.12.
'y F Ž u Ž x . . ;
x g Ž 0, x 1 .
Ž 3.4.13.
which gives u⬘ Ž x . s '2 and u⬘ Ž x . s y'2
'y F Ž u Ž x . . ;
x g Ž Ž 2 n q 1 . x 1 , 1 . . Ž 3.4.14.
120
ANURADHA ET AL.
FIGURE 3.11
Integrating Ž3.4.13. and Ž3.4.14. on Ž0, x 1 . and ŽŽ2 n q 1. x 1 , 1., respectively, we get ds
HuuŽ0.x 'y F Ž s . Ž .
s '2 x ;
x g Ž 0, x 1 .
Ž 3.4.15.
and ds
HuuŽ x1. 'y F Ž s . Ž .
s y'2 Ž 1 y x . ;
x g Ž Ž 2 n q 1 . x 1 , 1 . . Ž 3.4.16.
Substituting x s x 1 in Ž3.4.15. and x s Ž2 n q 1. x 1 in Ž3.4.16. and solving the resulting equations we get x1 s
H0 dsr
Ž 2 n q 1.
H0
ž 'y F Ž s . / dsr ž 'y F Ž s . / y H
m r␣
dsr
ž 'y F Ž s . /
Ž 3.4.17.
121
POSITIVE SOLUTIONS
and
'
s
2n q 1
'2
ds
H0 'y F Ž s .
q
1
ds
'2
Hmr␣ 'y F Ž s .
.
Ž 3.4.18.
Further, substituting x s 1 in Ž3.4.14., we get
'
s
m
.
'2 'y F Ž mr␣ .
Ž 3.4.19.
Combining Ž3.4.18. and Ž3.4.19., for such a solution to exist, there must exist an m such that ds
Ž 2 n q 1. H
0
'y F Ž s .
q
ds
Hmr␣ 'y F Ž s .
s
m
'y F Ž mr␣ .
. Ž 3.4.20.
We will investigate again, if such m exist. For m g Ž0, ␣ ., define
˜Ž m . s G
ds
Hmr␣ 'F Ž . y F Ž s . H Ž m. s
q Ž 2 n q 1.
ds
H0 'y F Ž s .
m
'F Ž . y F Ž mr␣ .
and
.
˜Ž m. is a decreasing function of m and H Ž m. is Then it is easy to see that G an increasing function of m where H Ž m. ª 0 as m ª 0, H Ž m. ª ⬁ as m ª ␣ . Then following a similar argument as in the case s one can ˜Ž mU2 nq1, 1 . s prove that there exists a unique mU2 nq1, 1 such that G U Ž . H m 2 nq1, 1 . By back-tracking, one can prove the existence of nonnegative solutions with 2 n q 1, n s 0, 1, 2, . . . interior critical points of Ž1.1. ᎐ Ž1.3. defined by Ž3.4.15. ᎐ Ž3.4.17. satisfying uŽ0. s 0, u⬘Ž1. s ymU2 nq1, 1 at s 2 nq1, 1 Ž0, mU2 nq1, 1 . given by Ž3.4.18. Žor Ž3.4.19... We now discuss a nonnegative solution satisfying uŽ0. s 0 s u⬘Ž1. with 2 n q 1 interior critical points, where n s 0, 1, 2, . . . Žsee Fig. 3.11b.. Here x 1 s 2Ž n 1q 1 . and the problem is equivalent to studying a nonnegative solution to yu⬙ Ž x . s f Ž u Ž x . . ; u Ž 0 . s 0 s u⬘ Ž x 1 .
x g Ž 0, x 1 .
Ž or Ž 0, 2 x 1 . . Ž or u Ž 0. s 0 s u Ž 2 x 1 . . .
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ANURADHA ET AL.
Again by the quadrature method, it easily follows that such a solution defined by ds uŽ x . s '2 x ; x g w 0, x 1 x , Ž 3.4.21. y F Ž s. 0
H '
exists at s 2 nq1, 2 Ž0, 0. where
'
2 nq1 , 2
Ž 0, 0 . s
2 Ž n q 1.
ds
H0 'y F Ž s .
'2
.
Ž 3.4.22.
We now summarize the above information on nonnegative solutions satisfying uŽ0. s 0 with 2 n q 1 interior critical points, where n s 0, 1, 2, . . . in the following theorem. THEOREM 3.4.1 Žb.. Let n s 0, 1, 2, . . . , ␣ g Ž0, ⬁. and let mU2 nq1, 1 be as described abo¨ e. For s 0, Ž1.1. ᎐ Ž1.3. ha¨ e exactly two nonnegati¨ e solutions u 2 nq1, i , i s 1, 2 with 2 n q 1 interior critical points at s 2 nq 1, 1 Ž0, mU2 nq 1, 1 . and s 2 nq 1, 2 Ž0, 0 ., respecti¨ ely. Here 2 nq1, 1Ž0, mU2 nq1, 1 . is gi¨ en by
'
s
2n q 1
'2
ds
H0 'y F Ž s .
q
1
ds
'2 Hmr␣ 'y F Ž s .
or
'
s
m
'2 'y F Ž mr␣ .
.
The corresponding solution is gi¨ en by ds
HuuŽ0.x 'y F Ž s . Ž .
s '2 x ;
x g Ž 0, x 1 .
and ds
HuuŽ x1. 'y F Ž s . Ž .
s y'2 Ž 1 y x . ;
x g Ž Ž 2 n q 1. x1 , 1.
where x 1 s x 1Ž mU2 nq1, 1 . is gi¨ en by x1 s
H0 dsr
Ž 2 n q 1 . H0
ž 'y F Ž s . / dsr ž 'y F Ž s . / y H
m r␣
dsr
ds
ž 'y F Ž s . /
Also 2 nq1, 2 Ž0, 0. is gi¨ en by
'
2 nq1, 2
Ž 0, 0 . s
2 Ž n q 1.
'2
H0 'y F Ž s .
.
123
POSITIVE SOLUTIONS
and the corresponding solution is defined by ds
H0u x 'y F Ž s . Ž .
s '2 x ;
x g w 0, x 1 x
where x 1 s 2Ž n 1q 1 . . Further note that 2 nq1, 1Ž0, mU2 nq1, 1 . - 2 nq1, 2 Ž0, 0. 2Ž nq1.q1, 1Ž0, mU2Ž nq1.q1, 1 .. Remark. Here we note that
2 n , 2 Ž , 0 . - 2 nq1, 1 Ž 0, mU2 nq1, 1 . 2 nq1, 1 Ž 0,
mU2 nq1, 1
. - 2Ž nq1., 1 Ž ,
and
mU2Ž nq1., 1
..
3.5. Ž , . Bifurcation Diagram In this section, given ) 0 and ␣ g Ž0, ⬁., we summarize our results on the existence of nonnegative solution for the semipositone case via the Ž , . bifurcation diagram ŽFig. 3.12. and Theorem 3.5.1. THEOREM 3.5.1. ŽA. Gi¨ en g Ž0, 0, 2 Ž , 0.. and ␣ g Ž0, ⬁., Ž1.1. ᎐ Ž1.3. ha¨ e at least one nonnegati¨ e solution with no interior critical points. ŽB. Gi¨ en n s 0, 1, 2, . . . , g Ž 2 n, 1Ž , mU2 n, 1 ., 2 n, 2 Ž , mU2 n, 2 .. and ␣ g Ž0, ⬁., Ž1.1. ᎐ Ž1.3. ha¨ e at least one nonnegati¨ e solution with 2 n interior critical points.
FIGURE 3.12
124
ANURADHA ET AL.
Gi¨ en n s 0, 1, 2, . . . , g Ž 2 nq1, 1Ž , mU2 nq1, 1 ., 2 nq1, 2 Ž , and ␣ g Ž0, ⬁., Ž1.1. ᎐ Ž1.3. ha¨ e at least one nonnegati¨ e solution with 2 n q 1 interior critical points. ŽC.
mU2 nq1, 2 ..
REFERENCES 1. V. Anuradha, C. Brown and R. Shivaji, Explosive nonnegative solutions to two point boundary value problems, Nonlinear Anal. 26, No. 3 Ž1996., 613᎐630. 2. V. Anuradha and R. Shivaji, A quadrature method for classes of multi-parameter two point boundary value problems, Appl. Anal. 54 Ž1994., 263᎐281. 3. V. Anuradha and R. Shivaji, Sign changing solutions for a class of superlinear multiparameter semipositone problems, Nonlinear Anal. 24, No. 11 Ž1995., 1581᎐1596. 4. A. Castro and R. Shivaji, Nonnegative solutions for a class of non-positone problems, Proc. Roy. Soc. Edinburgh, Sect. A 108 Ž1988., 291᎐302. 5. T. Laetsch, The number of solutions of a nonlinear two point boundary problems, Indiana Uni¨ . Math. J. 20 Ž1970r1971., 1᎐13. 6. A. R. Miciano and R. Shivaji, Multiple positive solutions for a class of semipositone Neumann two point boundary value problems, J. Math. Anal. Appl. 178 Ž1993., 102᎐115.
Positive Solutions for a Class of Nonlinear Boundary Value Problems with Neumann᎐Robin Boundary Conditions V. Anuradha 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 yu⬙ Ž x . s f Ž u Ž x . . ;
0-x-1
u⬘ Ž 0 . s 0; u⬘ Ž 1 . q ␣ u Ž 1 . s 0 where ) 0 and ␣ ) 0 are parameters, and f g C 2 w0, 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. 䊚 1999 Academic Press
1. INTRODUCTION Consider the nonlinear boundary value problem yu⬙ Ž x . s f Ž u Ž x . . ;
0-x-1
Ž 1.1.
u⬘ Ž 0 . s 0;
Ž 1.2.
u⬘ Ž 1 . q ␣ u Ž 1 . s 0
Ž 1.3.
94 0022-247Xr99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
95
POSITIVE SOLUTIONS
where ) 0 and ␣ ) 0 are parameters and f g C 2 w0, 1x. For positone problems, since f Ž s . ) 0 for s G 0, for the case ␣ s 0 Ž u⬘Ž0. s 0 s u⬘Ž1.., 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 analyzing 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 f Ž s.
lim
s
sª⬁
s G 0, s ⬁,
f ⬘ Ž s . G 0, f ⬙ Ž s . ) 0, F Ž s. s
s
H0 f Ž t . dt.
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 u⬘Ž x ., we have yu⬙ Ž x . u⬘Ž x . s f Ž uŽ x .. u⬘Ž x . and integrating the above equation, we obtain y
u⬘ Ž x . 2
2
s F Ž u Ž x . . q C.
Since uŽ0. s , C s y F Ž uŽ0.. s y F Ž . and therefore y
u⬘ Ž x . 2
2
s F Ž uŽ x . . y F Ž . ;
x g Ž 0, 1 .
Ž 2.1.
96
ANURADHA ET AL.
FIGURE 2.1
and u⬘ Ž x . s y'2
'F Ž . y F Ž u Ž x . . ;
x g Ž 0, 1 . .
Ž 2.2.
x g Ž 0, 1 . .
Ž 2.3.
Integrating Ž2.2. on Ž0, x ., we have ds
HuuŽ0.x 'F Ž . y F Ž s . Ž .
s y'2 x ;
Let u⬘Ž1. s ym where m ) 0. Then uŽ1. s in Ž2.3., we obtain
'
s
1
m ␣
g Ž0, .. Substituting x s 1
ds
Ž 2.4.
'2 Hmr␣ 'F Ž . y F Ž s .
and substituting x s 1 in Ž2.2., we get
'
s
m
'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
Hmr␣ 'F Ž . y F Ž s .
s
m
'F Ž . y F Ž mr␣ .
We first investigate whether such an m exists.
.
Ž 2.6.
97
POSITIVE SOLUTIONS
For m g Ž0, ␣ ., define GŽ m. s Hm r ␣ H0
ds
'F Ž . y F Ž s .
ds
'F Ž . y F Ž s .
) 0, GŽ ␣ . s 0, and G⬘Ž m. s
. Then GŽ0. s
y1
␣ F Ž . y F Ž mr ␣ .
'
. Hence,
for a given ␣ g Ž0, ⬁. and g Ž0, ⬁., GŽ m. is a decreasing function of m m Žsee Fig. 2.2.. Let H Ž m. s ; then H Ž0. s 0, H Ž m. ª ⬁ as
'F Ž . y F Ž mr␣ .
m ª ␣ , and H⬘ Ž m . s
2 ␣ F Ž . y F Ž mr␣ . q mf Ž mr␣ . 2 ␣ F Ž . y F Ž mr␣ .
3r2
) 0.
Hence, for a given ␣ g Ž0, ⬁. and g Ž0, ⬁., H Ž m. is an increasing function of m Žsee Fig. 2.2.. Thus, given ␣ g Ž0, ⬁. and g Ž0, ⬁., clearly there exists a unique m s m*Ž ␣ , . g Ž0, ␣ . such that GŽ m*. s H Ž 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, ⬁. and g Ž0, ⬁., there exists a unique m s m*Ž ␣ , . g Ž0, ␣ . such that ds
Hmr␣ 'F Ž . y F Ž s .
s
m
'F Ž . y F Ž mr␣ .
for m s m*. Also, there exists a unique s Ž , m*. gi¨ en by either
'
s
1
'2
ds
Hmr␣ 'F Ž . y F Ž s .
FIGURE 2.2
98
ANURADHA ET AL.
or
'
s
m
'2 'F Ž . y F Ž mr␣ .
for which Ž1.1. ᎐ Ž1.3. has a unique positi¨ e solution uŽ x . as defined by ds
HuuŽ0.x 'F Ž . y F Ž s . Ž .
s y'2 x ;
x g Ž 0, 1 .
with uŽ0. s and u⬘Ž1. s ym*. COROLLARY 2.2. Gi¨ en ␣ g Ž0, ⬁., the bifurcation diagram Ž , . of the positi¨ e solutions of Ž1.1. ᎐ Ž1.3. is described by either
'
s
1
'2
ds
Hmr␣ 'F Ž . y F Ž s .
or
'
s
m
'2 'F Ž . y F Ž mr␣ .
for m s m*. Since m* g Ž0, ␣ ., we have 0 - Ž , m*. - Ž . where 1
'2
H0
ds
'F Ž . y F Ž s .
' Ž .
s
. Note that s Ž . describes the bifurcation diagram
of positive solutions of the boundary value problem yu⬙ Ž x . s f Ž u Ž x . . ; u⬘ Ž 0 . s 0 s u Ž 1 .
0-x-1
Ž y1 - x - 1 .
Ž u Ž y1. . s 0 su Ž 1 . . .
f Ž s.
Since lim s ª⬁ s ⬁ it follows that lim ª⬁ Ž . s 0. Hence, given s Ž ) 0 and ␣ g 0, ⬁., 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, ⬁., Ž1.1. ᎐ Ž1.3. ha¨ e at least one positi¨ e solution with no interior critical points.
99
POSITIVE SOLUTIONS
FIGURE 2.3
3. EXISTENCE RESULTS WHEN f Ž0. - 0 Throughout this section we will assume the following: ᭚  , ) 0 such f s
Ž . that f Ž s . - 0 on w0,  ., f Ž  . s 0, f ⬘Ž s . G 0, f ⬙ Ž s . ) 0, lim s ª⬁ s ⬁, s s and F Ž . s 0 where F Ž s . s H0 f Ž t . 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 2 Ž , n. for which Ž1.1. ᎐ Ž1.3. have nonnegative solutions with n interior critical points. We denote these solutions by u n, 1Ž , 1 , ␣ . and u n, 2 Ž , 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 0, 4 then necessarily u⬘Ž1. - 0, while if g 0, 4 then for each n s 0, 1, 2, . . . , there exists a solution with u⬘Ž1. - 0 at s 1 and a second solution with u⬘Ž1. 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 u 0, 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 describing our results. 3.1. Existence Results when ) Let uŽ x . be a nonnegative solution of Ž1.1. ᎐ Ž1.3. with uŽ0. s g Ž , ⬁.. Since )  and f ) 0 for u )  , u must be initially concave down. Further, there does not exist 0 F r - such that F Ž . s F Ž 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
101
POSITIVE SOLUTIONS
Then building a quadrature method as was done in section 2, we get u⬘ Ž x .
y
2
s F Ž uŽ x . . y F Ž . ;
2
x g Ž 0, 1 .
Ž 3.1.1.
x g Ž 0, 1 . .
Ž 3.1.2.
and u⬘ Ž x . s y'2
'F Ž . y F Ž u Ž x . . ;
Integrating Ž3.1.2. on Ž0, x ., we have ds
HuuŽ0.x 'F Ž . y F Ž s . Ž .
s y'2 x ;
x g Ž 0, 1 . .
Ž 3.1.3.
Let u⬘Ž1. s ym where m ) 0. Then uŽ1. s mr␣ g Ž0, .. Substituting x s 1 in Ž3.1.3., we obtain
'
s
1
ds
Ž 3.1.4.
'2 Hmr␣ 'F Ž . y F Ž s .
and substituting x s 1 in Ž3.1.2., we get
'
s
m
'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
Hmr␣ 'F Ž . y F Ž s .
s
m
'F Ž . y F Ž mr␣ .
.
Ž 3.1.6.
We now investigate whether such an m exists. Given g Ž , ⬁., if we define GŽ m. s
ds
Hmr␣ 'F Ž . y F Ž s .
and
H Ž m. s
m
'F Ž . y F Ž mr␣ .
,
then GŽ m. is a decreasing function of m. Further, H Ž0. s 0, H Ž m. ª ⬁ as m ª ␣ and H⬘ Ž m . s
2 ␣ F Ž . y F Ž mr␣ . q mf Ž mr␣ . 2 ␣ F Ž . y F Ž mr␣ .
3r2
.
102
ANURADHA ET AL.
Note that H⬘Ž m. ) 0 for m G ␣ . For m g Ž0, ␣ ., define k Ž m. s 2 ␣ w F Ž . y F Ž mr␣ .x q mf Ž mr␣ .. Then k Ž0. s 2 ␣ F Ž . ) 0, k⬘Ž m. s yf Ž mr␣ . q m␣ f ⬘Ž mr␣ . ) 0 since f Ž s . - 0 on Ž0,  . and f ⬘Ž s . G 0. Thus, k Ž m. ) 0 on Ž0, ␣ . and hence H Ž m. is an increasing function of m. Thus GŽ m. and H Ž 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 H Ž 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, ⬁. and g Ž , ⬁., there exists a unique s mU0, 1Ž ␣ , . such that
mU0, 1
ds
Hmr␣ 'F Ž . y F Ž s .
s
m
'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
'
s
1
ds
'2 Hmr␣ 'F Ž . y F Ž s .
or
'
s
m
'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 ds
HuuŽ0.x 'F Ž . y F Ž s . Ž .
s y'2 x ;
with uŽ0. s and u⬘Ž1. s ymU0, 1.
FIGURE 3.3
x g Ž 0, 1 .
103
POSITIVE SOLUTIONS
COROLLARY 3.1.2. Gi¨ en ␣ g Ž0, ⬁., the bifurcation diagram Ž , . of the positi¨ e solutions of Ž1.1. ᎐ Ž1.3. with uŽ0. s ) is described by either
'
s
1
ds
'2 Hmr␣ 'F Ž . y F Ž s .
or
'
s
m
'2 'F Ž . y F Ž mr␣ .
for m s mU0, 1. Since mU0, 1 g Ž0, ␣ ., we have 0 - 0, 1Ž , mU0, 1 . - Ž . where s
' Ž . s '12 H
0
ds
'F Ž . y F Ž s .
describes the bifurcation diagram of positive
solutions of the problem yu⬙ Ž u . s f Ž u Ž x . . ; u⬘ Ž 0 . s 0 s u Ž 1 .
0-x-1
Ž y1 - x - 1 .
Ž u Ž y1. . s 0 su Ž 1 . . .
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 yu⬙ Ž x . s f Ž u Ž x . . ; u⬘ Ž 0 . s 0,
0-x-1
u Ž 0. 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 u⬘Ž1. s y␣ uŽ1. - 0. ŽFor if uŽ1. s 0 then u⬘Ž1. s 0 and hence by Ž3.2.1.. Žbelow. we get F Ž . s F Ž0. s 0, which is a contradiction to the fact that F Ž . / 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, x 0 x and w2 nx 0 , 1x, where x 0 is the first interior critical point. Let uŽ0. s uŽ2 x 0 . s ⭈⭈⭈ s uŽ2 nx 0 . s g Ž  , .. Let r g Ž0,  . be the unique number such that F Ž . s F Ž r . Žsee Fig. 3.5.. Then uŽ x 0 . s uŽ3 x 0 . s ⭈⭈⭈ s uŽŽ2 n y 1. x 0 .
104
ANURADHA ET AL.
FIGURE 3.4
FIGURE 3.5
s r. Thus we see that a nonnegative solution of Ž1.1. ᎐ Ž1.3. with uŽ0. s g Ž  , . and 2 n interior critical points, where n s 0, 1, 2, . . . , is positive in w0, 1x. Building a quadrature method as before we see that
y
u⬘ Ž x . 2
2
s F Ž uŽ x . . y F Ž . ;
x g Ž 0, 1 .
Ž 3.2.1.
x g Ž 0, x 0 .
Ž 3.2.2.
which gives u⬘ Ž x . s y'2
'F Ž . y F Ž u Ž x . . ;
105
POSITIVE SOLUTIONS
and u⬘ Ž x . s y'2
'F Ž . y F Ž u Ž x . . ;
x g Ž 2 nx 0 , 1 . . Ž 3.2.3.
Integrating Ž3.2.2. and Ž3.2.3. on Ž0, x. and Ž2 nx 0 , 1. respectively, we get ds
HuuŽ0.x 'F Ž . y F Ž s . Ž .
s y'2 x ;
x g Ž 0, x 0 .
Ž 3.2.4.
and ds
HuuŽ x1. 'F Ž . y F Ž s . Ž .
s y'2 Ž 1 y x . ;
x g Ž 2 nx 0 , 1 . . Ž 3.2.5.
Let u⬘Ž1. s ym where m ) 0. Then uŽ1. s m␣ g Ž r, .. Substituting x s x 0 in Ž3.2.4. and x s 2 nx 0 in Ž3.2.5. and solving the resulting equations we get x0 s
ž 'F Ž . y F Ž s . / dsr ž 'F Ž . y F Ž s . / q H dsr ž 'F Ž . y F Ž s . / Hr dsr
2 nHr
r
m r␣
Ž 3.2.6. and
'
s
2n
ds
'2 Hr 'F Ž . y F Ž s .
q
1
ds
'2 Hmr␣ 'F Ž . y F Ž s .
. Ž 3.2.7.
Further, substituting x s in Ž3.2.3., we get
'
s
m
'2 'F Ž . y F Ž mr␣ .
.
Ž 3.2.8.
Combining Ž3.2.7. and Ž3.2.8., for such a solution to exist, there must exist an m such that 2n
ds
Hr 'F Ž . y F Ž s .
q
ds
Hmr␣ 'F Ž . y F Ž s .
s
m
'F Ž . y F Ž mr␣ .
.
Ž 3.2.9. We now investigate, as before, if such an m exists.
106
ANURADHA ET AL.
For m g Ž ␣ r, ␣ ., define GŽ m. s
ds
Hmr␣ 'F Ž . y F Ž s . H Ž m. s
ds
q 2n
and
Hr 'F Ž . y F Ž s .
m
'F Ž . y F Ž mr␣ .
.
Then GŽ m. is a decreasing function of m. Now H Ž m. ª ⬁ as m ª ␣ r, ␣ and H⬘ Ž m . s
2 ␣ F Ž . y F Ž mr␣ . q mf Ž mr␣ . 2 ␣ F Ž . y F Ž mr␣ .
3r2
.
Then H⬘Ž m. ) 0 for m G ␣ . For m g Ž ␣ r, ␣ . define hŽ m. s 2 ␣ w F Ž . y F Ž mr␣ .x q mf Ž mr␣ .. Then hŽ ␣ r . s ␣ rf Ž r . - 0 Žsince r -  ., hŽ ␣ . s 2 ␣ w F Ž . y F Ž  .x ) 0 and h⬘Ž m. s yf Ž mr␣ . q m Ž . Ž . Ž . Ž . ␣ f ⬘ mr␣ ) 0 since f s - 0 on r,  and f ⬘ s G 0. Thus there exists a unique c g Ž ␣ r, ␣ . such that hŽ c . s 0. Thus H Ž m. and GŽ m. have one of the forms described in Fig. 3.6. Now, to investigate whether H Ž m. and GŽ m. intersect, we define LŽ m. s H Ž m. y GŽ m. Žsee Fig. 3.7.. That is, LŽ m. s
m
'F Ž . y F Ž mr␣ . y 2n
ds
y
Hr 'F Ž . y F Ž s .
ds
Hmr␣ 'F Ž . y F Ž s . .
Then LŽ m. ª ⬁ as m ª ␣ r, ␣ and L⬘ Ž m . s
2 Ž ␣ q 1 . F Ž . y F Ž mr␣ . q mf Ž mr␣ . 2 ␣ F Ž . y F Ž mr␣ .
3r2
.
It is easy to see that L⬘Ž m. ) 0 on w ␣ , ␣ .. For m g Ž ␣ r, ␣ . define k Ž m. s 2Ž ␣ q 1.w F Ž . y F Ž mr␣ .x q mf Ž mr␣ .. Then k Ž ␣ r . s ␣ rf Ž r . 0 Žsince r -  ., k Ž ␣ . s 2Ž ␣ q 1.w F Ž . y F Ž  .x ) 0, and k⬘Ž m. s ␣q 2 Žyf Ž mr␣ .. q Ž mr␣ . f ⬘Ž mr␣ . ) 0. Thus there exists a unique c g ␣ Ž ␣ r, ␣ . such that k Ž c . s 0, i.e., L⬘Ž c . s 0. Note that Ž1.1. ᎐ Ž1.3. have no positive solution if LŽ c . ) 0. Since k Ž c . s 0, we have F Ž . y F Ž ␣c . s
POSITIVE SOLUTIONS
FIGURE 3.6
FIGURE 3.7
107
108 y
ANURADHA ET AL.
c 2Ž ␣ q 1.
f Ž ␣c .. So LŽ c . s
c
'F Ž . y F Ž cr␣ . y 2n
s
y
(
ds
Hcr␣ 'F Ž . y F Ž s .
ds
Hr 'F Ž . y F Ž s .
2 c Ž ␣ q 1. yf Ž cr␣ .
y
ds
y 2n
ds
Hcr␣ 'F Ž . y F Ž s .
Hr 'F Ž . y F Ž s .
.
Since c g Ž ␣ r, ␣ ., we have
(
2 c Ž ␣ q 1. yf Ž cr␣ .
y Ž 2 n q 1.
ds
Hr 'F Ž . y F Ž s .
F LŽ c . F
(
2 c Ž ␣ q 1. yf Ž cr␣ .
y
ds
y 2n
H 'F Ž . y F Ž s .
ds
Hr 'F Ž . y F Ž s .
.
Ž 3.2.10. Using the right inequality of Ž3.2.10. we see that LŽ c . F
F
F
( ( (
2 c Ž ␣ q 1. yf Ž cr␣ . 2 c Ž ␣ q 1. yf Ž cr␣ . 2 c Ž ␣ q 1. yf Ž cr␣ .
ds
y Ž 2 n q 1.
H 'F Ž . y F Ž s .
y Ž 2 n q 1.
H 'F Ž . y F Ž  .
y
ds
Ž 2 n q 1. Ž y  . 'F Ž . y F Ž  . .
We see that as ª , r ª 0, and k Ž ␣ r . s ␣ rf Ž r . ª 0y and so c ª 0. Thus lim ª LŽ c . G y
Ž2 n q 1 . Ž y  .
'y F Ž  .
- 0. Similarly, using the left in-
109
POSITIVE SOLUTIONS
equality of Ž3.2.10. we get
LŽ c . G
(
2 c Ž ␣ q 1. yf Ž cr␣ .
y Ž 2 n q 1.
ds
Hr 'F Ž . y F Ž s .
.
But as ª  , r ª  and c ª ␣ . Thus lim ª  LŽ c . s ⬁. Hence we see that LŽ c . - 0 for close to and LŽ c . ) 0 for close to  . Therefore, there exists an a*n g Ž  , . such that for each g Ž aUn , ., LŽ cŽ .. - 0. Therefore, for each g Ž aUn , ., there exists mU2 n, 1 g Ž cŽ ., ␣ . and mU2 n, 2 g Ž ␣ r, cŽ .. such that LŽ mU2 n, 1 . s 0 s LŽ mU2 n, 2 .. For s aUn we have LŽ cŽ aUn .. s 0 and so there exists a unique mU2 n, 1 g Ž ␣ r, ␣ . such that LŽ mU2 n, 1 . s 0. Further there exists bnU g Ž  , aUn x such that for g w  , bnU . there are no m’s such that LŽ m. s 0. Note also that aUn ) aUnq1 for each n s 0, 1, 2, . . . . Now, by back-tracking one can prove the existence of such solutions of Ž1.1. ᎐ Ž1.3. as described in the Theorem 3.2.1 below. THEOREM 3.2.1. Let n s 0, 1, 2, . . . , ␣ g Ž0, ⬁. and let mU2 n, 1 , mU2 n, 2 , bnU be as described abo¨ e.
aUn ,
ŽA. Then for g Ž aUn , . Ž1.1. ᎐ Ž1.3. ha¨ e exactly two solutions u 2 n, i , i s 1, 2 with 2 n interior critical points at s 2 n, i Ž , mU2 n, i ., i s 1, 2 respecti¨ ely. Here 2 n, i Ž , mU2 n, 1 ., i s 1, 2 are gi¨ en by
'
s
2n
ds
'2 Hr 'F Ž . y F Ž s .
q
1
ds
'2 Hmr␣ 'F Ž . y F Ž s .
or
'
s
m
'2 'F Ž . y F Ž mr␣ .
.
The corresponding solutions are defined by ds
HuuŽ0.x 'F Ž . y F Ž s . Ž .
ds
HuuŽ x1. 'F Ž . y F Ž s . Ž .
s y'2 x ;
s y'2 Ž 1 y x . ;
x g Ž 0, x 0 . , x g Ž 2 nx 0 , 1 .
110
ANURADHA ET AL.
where x 0 s x 0 Ž , mU2 n, i . is gi¨ en by x0 s
ž 'F Ž . y F Ž s . / dsr ž 'F Ž . y F Ž s . / q H dsr ž 'F Ž . y F Ž s . / Hr dsr
2 nHr
r
m r␣
.
Further, 2 n, 1Ž , mU2 n, 1 . - 2 n, 2 Ž , mU2 n, 2 .. ŽB. For s aUn , Ž1.1. ᎐ Ž1.3. ha¨ e exactly one solution u 2 n, 1 with 2 n interior critical points at s 2 n, 1Ž aUn , mU2 n, 1 . where 2 n, 1 and u 2 n, 1 are as defined in ŽA.. ŽC. For g w  , bnU ., Ž1.1. ᎐ Ž1.3. ha¨ e no solution with 2 n interior critical points for any . COROLLARY 3.2.2. Gi¨ en ␣ g Ž0, ⬁., the bifurcation diagram Ž , . of positi¨ e solutions with 2 n interior critical point of Ž1.1. ᎐ Ž1.3. with uŽ0. s g w  , . is described by
'
s
2n
'2
ds
Hr 'F Ž . y F Ž s .
q
1
'2
ds
Hmr␣ 'F Ž . y F Ž s .
or
'
s
m
'2 'F Ž . y F Ž mr␣ .
for m s mU2 n, 1 and m s mU2 n, 2 . 3.3. Existence Results when g Ž0,  . First we note that if u is a nonnegative solution of Ž1.1. ᎐ Ž1.3. with Ž u 0. s g Ž0,  ., u must have at least one critical point Žsince u⬘Ž1. - 0.. In fact, u must have an odd number of interior critical points. In general, we consider a nonnegative solution with 2 n q 1, n s 0, 1, 2, . . . interior critical points Žsee Fig. 3.8.. Again, since Ž1.1. is autonomous, it is enough to study a solution in w0, x 1 x and wŽ2 n q 1. x 1 , 1x where x 1 is the first interior critical point of u. Suppose uŽ0. s uŽ2 x 1 . s ⭈⭈⭈ s uŽ2 nx 1 . s g Ž0,  .. Let r g Ž  , . be such that F Ž . s F Ž r .. Then uŽ x 1 . s uŽ3 x 1 . s ⭈⭈⭈ s uŽŽ2 n q 1. x 1 . s r. In this case also, all nonnegative solutions are positive in w0, 1x. Building a quadrature method as before, we see that y
u⬘ Ž x . 2
2
s F Ž uŽ x . . y F Ž . ;
x g Ž 0, 1 .
Ž 3.3.1.
111
POSITIVE SOLUTIONS
FIGURE 3.8
which gives u⬘ Ž x . s '2
'F Ž . y F Ž u Ž x . . ;
x g Ž 0, x 1 .
Ž 3.3.2.
and u⬘ Ž x . s y'2
'F Ž . y F Ž u Ž x . . ;
x g Ž Ž 2 n q 1 . x 1 , 1 . . Ž 3.3.3.
Integrating Ž3.3.2. and Ž3.3.3. on Ž0, x 1 . and ŽŽ2 n q 1. x 1 , 1., respectively, we get ds
HuuŽ0.x 'F Ž . y F Ž s . Ž .
s '2 x ;
x g Ž 0, x 1 .
Ž 3.3.4.
and ds
HuuŽ x1. 'F Ž . y F Ž s . Ž .
s y'2 Ž 1 y x . ;
x g Ž Ž 2 n q 1. x1 , 1. .
Ž 3.3.5. Let u⬘Ž1. s ym where m ) 0. Then uŽ1. s m␣ g Ž , r .. Substituting x s x 1 in Ž3.3.4. and x s Ž2 n q 1. x 1 in Ž3.3.5. and solving the resulting equations we get x1 s
ž 'F Ž . y F Ž x . / dsr ž 'F Ž . y F Ž s . / y H dsr ž 'F Ž . y F Ž s . / Hr dsr
Ž 2 n q 1 . Hr
r
m r␣
Ž 3.3.6.
112
ANURADHA ET AL.
and
'
s
Ž 2 n q 1.
'2
ds
r
q
H 'F Ž . y F Ž s .
1
ds
r
'2
Hmr␣ 'F Ž . y F Ž s .
.
Ž 3.3.7. Further, substituting x s 1 in Ž3.3.3., we get
'
m
s
'F Ž . y F Ž mr␣ .
.
Ž 3.3.8.
Combining Ž3.3.7. and Ž3.3.8., for such a solution to exist, there must exist an m such that
Ž 2 n q 1. H
s
ds
r
'F Ž . y F Ž s . m
'F Ž . y F Ž mr␣ .
q
ds
Hmr␣ 'F Ž . y F Ž s .
.
Ž 3.3.9.
We now investigate, as before, if such an m exists. For m g Ž ␣ , ␣ r ., define GŽ m. s
r
ds
Hmr␣ 'F Ž . y F Ž s . H Ž m. s
q Ž 2 n q 1.
ds
r
H 'F Ž . y F Ž s .
m
'F Ž . y F Ž mr␣ .
and
.
Then following the similar argument as in Section 3.2 one can see that GŽ m. and H Ž m. have the form as in Fig. 3.4 Žwith r and interchanged.. To investigate whether GŽ m. and H Ž m. intersect, we define P Ž m. s H Ž m. y GŽ m.. That is, P Ž m. s
m
'F Ž . y F Ž mr␣ . y Ž 2 n q 1.
r
y ds
r
ds
Hmr␣ 'F Ž . y F Ž s .
H 'F Ž . y F Ž s .
113
POSITIVE SOLUTIONS
for m g Ž ␣ , ␣ r .. Then P Ž m. ª ⬁ as m ª ␣ , ␣ r and P⬘ Ž m . s
2 Ž ␣ q 1 . F Ž . y F Ž mr␣ . q mf Ž mr␣ . 3r2
2 ␣ F Ž . y F Ž mr␣ .
s L⬘ Ž m . .
Using a similar argument as in Section 3.2, one can see that there exists a unique c g Ž ␣ , ␣ . such that P⬘Ž c . s 0 s L⬘Ž c .. Now r c ds P Ž c. s q F Ž . y F Ž cr␣ . cr ␣ F Ž . y F Ž s .
H '
'
y Ž 2 n q 1.
ds
r
H 'F Ž . y F Ž s .
.
Since c g Ž ␣ , ␣ ., we have
(
2 c Ž ␣ q 1. yf Ž cr␣ .
y Ž 2 n q 1.
F P Ž c. F
(
2 c Ž ␣ q 1. yf Ž cr␣ .
y Ž 2 n q 1.
ds
r
H 'F Ž . y F Ž s .
r
y
ds
r
H 'F Ž . y F Ž s . ds
H 'F Ž . y F Ž s .
.
As before, we can show that lim ª 0 P Ž m. s lim r ª P Ž m. - 0 and lim ª  P Ž m. s lim r ª  P Ž m. s ⬁. Therefore there exists an a ˜n g Ž0,  . Ž . Ž Ž .. such that for each g 0, a ˜n , P c - 0. Thus for each g Ž0, a˜n ., U U there exists m 2 nq1, 1 s m 2 nq1, 1Ž ␣ , . g Ž cŽ ., ␣ r . and mU2 ny1, 2 s mU2 nq1, 1Ž ␣ , . g Ž ␣ , cŽ .. such that P Ž mU2 nq1, 1 . s 0 s P Ž mU2 nq1, 2 .. For s a˜n , we have LŽ cŽ a˜n .. s 0 and so there exists a unique mU2 nq1, 1 g Ž ␣ , ␣ r . such that P Ž mU2 nq1, 1 . s 0. Further, there exists ˜ bn g w a ˜n ,  . ˜ Ž . Ž . such that for g bn ,  there are no m’s such that P m s 0. Note again that a ˜n - a˜nq1 for each n s 0, 1, 2, . . . . Now, by back-tracking one can prove the existence of such solutions of Ž1.1. ᎐ Ž1.3. as described in Theorem 3.3.1 below. THEOREM 3.3.1. Let n s 0, 1, 2, . . . , ␣ g Ž0, ⬁. and let mU2 nq1, 1 , mU2 nq1, 2 , a ˜n , ˜bn be as described abo¨ e. ŽA. Then for g Ž0, a ˜n . Ž1.1. ᎐ Ž1.3. ha¨ e exactly two solutions u 2 nq1, i , i s 1, 2 with 2 n q 1 interior critical points at s 2 nq1, i Ž , mU2 nq1, i ., i s 1, 2 respecti¨ ely. Here 2 n, i Ž , mU2 n, i ., i s 1, 2 are gi¨ en by 2n q 1 r ds 1 r ds ' s q '2 F Ž . y F Ž s . '2 mr␣ F Ž . y F Ž s .
H'
H '
114
ANURADHA ET AL.
or
'
s
m
'2 'F Ž . y F Ž mr␣ .
.
The corresponding solutions are defined by ds
HuuŽ0.x 'F Ž . y F Ž s . Ž .
s '2 x ;
x g Ž 0, x 1 .
and ds
HuuŽ x1. 'F Ž . y F Ž s . Ž .
s y'2 Ž 1 y x . ;
x g Ž Ž 2 n q 1. x1 , 1.
where x 1 s x 1Ž , mU2 nq1, i . is gi¨ en by x1 s
ž 'F Ž . yF Ž s . / dsr ž 'F Ž . yF Ž s . / yH dsr ž 'F Ž . yF Ž s . / Hr dsr
Ž 2 nq1 . Hr
r
m r␣
.
Further, 2 nq1, 1Ž , mU2 nq1, 1 . - 2 nq1, 2 Ž , mU2 nq1, 2 .. ŽB. For s a ˜n , Ž1.1. ᎐ Ž1.3. ha¨ e exactly one solution u 2 nq1, 1 with 2 n q 1 interior critical points at s 2 nq1, 1Ž a ˜n , mU2 nq1, 1 . where 2 nq1, 1 and u 2 nq1, 1 are as defined in ŽA.. ŽC. For g Ž ˜ bn ,  ., Ž1.1. ᎐ Ž1.3. ha¨ e no solution with 2 n q 1 interior critical points for any . COROLLARY 3.2.2. Gi¨ en ␣ g Ž0, ⬁., the bifurcation diagram Ž , . of positi¨ e solutions with 2 n q 1 interior critical points of Ž1.1. ᎐ Ž1.3. with uŽ0. s p⑀ Ž0,  . is described by
'
s
2n q 1
'2
ds
r
H 'F Ž . y F Ž s .
q
1
r
or
'
s
m
'F Ž . y F Ž mr␣ .
for m s mU2 nq1, 1 and m s mU2 nq1, 2 .
ds
'2 Hmr␣ 'F Ž . y F Ž s .
115
POSITIVE SOLUTIONS
3.4. Existence Results when s and s 0 First we will study the case s . Let u be a nonnegatiive solution of Ž1.1. ᎐ Ž1.3. with uŽ0. s . Since uŽ0. s )  , u is initially concave down. Note that if uŽ1. s 0 then u⬘Ž1. s 0 and if uŽ1. / 0 and then u⬘Ž1. s ym - 0. Hence u can have either none or an even number of interior critical points. We will first discuss the case when u⬘Ž1. s ym - 0 with 2 n interior critical points, where n s 0, 1, 2, . . . ŽFig. 3.9a.. Since Ž1.1. is autonomous, as in Sections 3.2 and 3.3, it is enough to study the solution in w0, x 0 x and w2 nx 0 , 1x where x 0 is the first interior critical point of u. Let uŽ0. s uŽ2 x 0 . s ⭈⭈⭈ s uŽ2 nx 0 . s . Then uŽ x 0 . s uŽ3 x 0 . s ⭈⭈⭈ s uŽŽ2 n y 1. x 0 . s 0. Again building a quadrature method, we see that
y
u⬘ Ž x . 2
2
s F Ž uŽ x . . ;
FIGURE 3.9
x g Ž 0, 1 .
Ž 3.4.1.
116
ANURADHA ET AL.
which gives u⬘ Ž x . s y 2 F Ž u Ž x . . ;
'
x g Ž 0, x 0 .
Ž 3.4.2.
u⬘ Ž x . s y 2 F Ž u Ž x . . ;
x g Ž 2 nx 0 , 1 . .
Ž 3.4.3.
and
'
Integrating Ž3.4.2. and Ž3.4.3. on Ž0, x 0 . and Ž2 nx 0 , 1., respectively, we get ds
HuuŽ0.x 'y F Ž s . Ž .
s y'2 x ;
x g Ž 0, x 0 .
Ž 3.4.4.
and ds
HuuŽ x1. 'y F Ž s . Ž .
s y'2 Ž 1 y x . ;
x g Ž 2 nx 0 , 1 . .
Ž 3.4.5.
Let u⬘Ž1. s ym where m ) 0. Then uŽ1. s m␣ g Ž0, .. Substituting x s x 0 in Ž3.4.4. and x s 2 nx 0 in Ž3.4.5. and solving the resulting equations we get H0 dsr
x0 s
2 nH0
dsr
ž 'y F Ž s . /
ž 'y F Ž s . / q Hm r ␣ dsr
Ž 3.4.6.
ž 'y F Ž s . /
and
'
s
2n
'2
ds
H0 'y F Ž s .
q
1
'2
ds
Hmr␣ 'y F Ž s .
.
Ž 3.4.7.
Further, substituting x s 1 in Ž3.4.3., we get
'
s
m
'2 'y F Ž mr␣ .
.
Ž 3.4.8.
Combining Ž3.4.7. and Ž3.4.8., for such a solution to exist, there must exist an m such that 2n
ds
H0 'y F Ž s .
q
ds
Hmr␣ 'y F Ž s .
s
m
'y F Ž mr␣ .
We now investigate, as before, if such an m exists.
.
Ž 3.4.9.
117
POSITIVE SOLUTIONS
For m g Ž0, ␣ ., define GŽ m. s
ds
Hmr␣ 'y F Ž s . H Ž m. s
q 2n
ds
and
H0 'y F Ž s .
m
'2 'y F Ž mr␣ .
.
Then GŽ m. is a decreasing function of m. Now H Ž0. s 0, H Ž m. ª ⬁ as m ª ␣ and H⬘ Ž m . s
2 ␣ yF Ž mr␣ . q mf Ž mr␣ . 2 ␣ yF Ž mr␣ .
3r2
.
Then H⬘Ž m. ) 0 for m G ␣ . For m g Ž0, ␣ . define hŽ m. s y2 ␣ w F Ž mr␣ .x q mf Ž mr␣ .. Then hŽ0. s 0 and hŽ ␣ . s y2 ␣ F Ž  . ) 0 and h⬘Ž m. s yf Ž mr␣ . q m␣ ) 0 since f Ž s . - 0 on Ž0,  . and f ⬘Ž2. G 0. Thus H Ž m. is an increasing function of m Žsee Fig. Ž3.10. and hence there exists a unique mU2 n, 1 s mU2 n, 1Ž ␣ , . g Ž0, ␣ . such that GŽ mU2 n, 1 . s H Ž mU2 n, 1 .. Now, by back-tracking one can prove the existence of nonnegative solutions with 2 n, n s 0, 1, 2, . . . interior critical points of Ž1.1. ᎐ Ž1.3. defined by Ž3.4.4. ᎐ Ž3.4.6. with uŽ0. s , u⬘Ž1. s ymU2 n, 1 at s 2 n, 1Ž , mU2 n, 1 . given by Ž3.4.7. Žor Ž3.4.8... We next discuss a solution satisfying uŽ0. s and uŽ1. s u⬘Ž1. s 0 with 2 n interior critical points, where n s 0, 1, 2, . . . Žsee Fig. 3.9b.. Here x 0 s 1rŽ2 n q 1. and the problem is equivalent to studying a positive
FIGURE 3.10
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ANURADHA ET AL.
solution to yu⬙ Ž x . s f Ž u Ž x . . ;
x g Ž 0, x 0 .
Ž or Ž yx 0 , x 0 . . Ž or u Ž yx 0 . s 0 s u Ž x 0 . . .
u⬘ Ž 0 . s 0 s u Ž x 0 .
Here the quadrature method yields explicit relationships and it follows easily that such a solution defined by ds
H0u x 'y F Ž s . Ž .
s y'2 x ;
x g Ž 0, x 0 .
Ž 3.4.10.
exists at s 2 n, 2 Ž , 0., where
'
2 n, 2
Ž , 0. s
2n q 1
'2
ds
H0 'y F Ž s .
.
Ž 3.4.11.
Now we summarize the above information on solutions with 2 n, n s 0, 1, 2, . . . interior critical points in the following theorem. THEOREM 3.4.1 Ža.. Let n s 0, 1, 2, . . . , ␣ g Ž0, ⬁. and let mU2 n, 1 be as described abo¨ e. For s , Ž1.1. ᎐ Ž1.3. ha¨ e exactly two nonnegati¨ e solutions u 2 n, i , i s 1, 2 with 2 n interior critical points at s 2 n, 1Ž , mU2 n, 1 . and s 2 n, 2 Ž , 0., respecti¨ ely. Here 2 n, 1Ž , mU2 n, 1 . is gi¨ en by
'
s
2n
ds
'2 H0 'y F Ž s .
q
1
ds
'2 Hmr␣ 'y F Ž s .
or
'
s
m
'2 'y F Ž mr␣ .
.
The corresponding solution is gi¨ en by ds
HuuŽ0.x 'y F Ž s . Ž .
u Ž1 .
H
uŽ x .
ds
'y F Ž s .
s y'2 x ;
s y'2 Ž 1 y x . ;
x g Ž 0, x 0 . ,
x g Ž 2 nx 0 , 1 .
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POSITIVE SOLUTIONS
where x 0 s x 0 Ž , mU2 n, 1 . is gi¨ en by x0 s
H0 dsr 2 nH0
dsr
ž 'y F Ž s . /
ž 'y F Ž s . / q Hm r ␣ dsr
ž 'y F Ž s . /
.
Also 2 n, 2 Ž , 0. is gi¨ en by
'
2 n, 2
Ž , 0. s
2n q 1
'2
0
ds
H 'y F Ž s .
and the corresponding solution is defined by ds
H0u x 'y F Ž s . Ž .
s y'2 x ;
x g Ž 0, x 0 .
where x 0 s 1rŽ2 n q 1.. Further note that 2 n, 1Ž , mU2 n, 1 . - 2 n, 2 Ž , 0. 2Ž nq1., 1Ž ,mU2Ž nq1., 1 .. Now suppose u is a nonnegative solution of Ž1.1. ᎐ Ž1.3. with uŽ0. s 0. Note that if uŽ1. s 0, u⬘Ž1. s 0 and if u⬘Ž1. / 0 the u⬘Ž1. - 0. Thus u must have an odd number of interior critical points in either case. Hence the solutions are of forms as described in Fig. 3.11. We will first discuss the case when u⬘Ž1. s ym - 0 with 2 n q 1 interior critical points, where n s 0, 1, 2, . . . Žsee Fig. 3.11a.. Since u⬘Ž1. s ym - 0, uŽ1. s mr␣ . Note that in order to study a nonnegative solution with 2 n q 1 interior critical points, due to the symmetry it is enough to study the solution in the intervals w0, x 1 x and wŽ2 n q 1. x 1 , 1x, where x 1 is the first interior critical point of u. Suppose uŽ0. s uŽ2 x 1 . s uŽ4 x 1 . s ⭈⭈⭈ s uŽ2 nx 1 . s 0. Then uŽ x 1 . s uŽ3 x 1 . s ⭈⭈⭈ s uŽŽ2 n q 1. x 1 . s . Building a quadrature method as before, we get y
u⬘ Ž x .
2
2
s F Ž uŽ x . . ;
x g Ž 0, 1 .
Ž 3.4.12.
'y F Ž u Ž x . . ;
x g Ž 0, x 1 .
Ž 3.4.13.
which gives u⬘ Ž x . s '2 and u⬘ Ž x . s y'2
'y F Ž u Ž x . . ;
x g Ž Ž 2 n q 1 . x 1 , 1 . . Ž 3.4.14.
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ANURADHA ET AL.
FIGURE 3.11
Integrating Ž3.4.13. and Ž3.4.14. on Ž0, x 1 . and ŽŽ2 n q 1. x 1 , 1., respectively, we get ds
HuuŽ0.x 'y F Ž s . Ž .
s '2 x ;
x g Ž 0, x 1 .
Ž 3.4.15.
and ds
HuuŽ x1. 'y F Ž s . Ž .
s y'2 Ž 1 y x . ;
x g Ž Ž 2 n q 1 . x 1 , 1 . . Ž 3.4.16.
Substituting x s x 1 in Ž3.4.15. and x s Ž2 n q 1. x 1 in Ž3.4.16. and solving the resulting equations we get x1 s
H0 dsr
Ž 2 n q 1.
H0
ž 'y F Ž s . / dsr ž 'y F Ž s . / y H
m r␣
dsr
ž 'y F Ž s . /
Ž 3.4.17.
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POSITIVE SOLUTIONS
and
'
s
2n q 1
'2
ds
H0 'y F Ž s .
q
1
ds
'2
Hmr␣ 'y F Ž s .
.
Ž 3.4.18.
Further, substituting x s 1 in Ž3.4.14., we get
'
s
m
.
'2 'y F Ž mr␣ .
Ž 3.4.19.
Combining Ž3.4.18. and Ž3.4.19., for such a solution to exist, there must exist an m such that ds
Ž 2 n q 1. H
0
'y F Ž s .
q
ds
Hmr␣ 'y F Ž s .
s
m
'y F Ž mr␣ .
. Ž 3.4.20.
We will investigate again, if such m exist. For m g Ž0, ␣ ., define
˜Ž m . s G
ds
Hmr␣ 'F Ž . y F Ž s . H Ž m. s
q Ž 2 n q 1.
ds
H0 'y F Ž s .
m
'F Ž . y F Ž mr␣ .
and
.
˜Ž m. is a decreasing function of m and H Ž m. is Then it is easy to see that G an increasing function of m where H Ž m. ª 0 as m ª 0, H Ž m. ª ⬁ as m ª ␣ . Then following a similar argument as in the case s one can ˜Ž mU2 nq1, 1 . s prove that there exists a unique mU2 nq1, 1 such that G U Ž . H m 2 nq1, 1 . By back-tracking, one can prove the existence of nonnegative solutions with 2 n q 1, n s 0, 1, 2, . . . interior critical points of Ž1.1. ᎐ Ž1.3. defined by Ž3.4.15. ᎐ Ž3.4.17. satisfying uŽ0. s 0, u⬘Ž1. s ymU2 nq1, 1 at s 2 nq1, 1 Ž0, mU2 nq1, 1 . given by Ž3.4.18. Žor Ž3.4.19... We now discuss a nonnegative solution satisfying uŽ0. s 0 s u⬘Ž1. with 2 n q 1 interior critical points, where n s 0, 1, 2, . . . Žsee Fig. 3.11b.. Here x 1 s 2Ž n 1q 1 . and the problem is equivalent to studying a nonnegative solution to yu⬙ Ž x . s f Ž u Ž x . . ; u Ž 0 . s 0 s u⬘ Ž x 1 .
x g Ž 0, x 1 .
Ž or Ž 0, 2 x 1 . . Ž or u Ž 0. s 0 s u Ž 2 x 1 . . .
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Again by the quadrature method, it easily follows that such a solution defined by ds uŽ x . s '2 x ; x g w 0, x 1 x , Ž 3.4.21. y F Ž s. 0
H '
exists at s 2 nq1, 2 Ž0, 0. where
'
2 nq1 , 2
Ž 0, 0 . s
2 Ž n q 1.
ds
H0 'y F Ž s .
'2
.
Ž 3.4.22.
We now summarize the above information on nonnegative solutions satisfying uŽ0. s 0 with 2 n q 1 interior critical points, where n s 0, 1, 2, . . . in the following theorem. THEOREM 3.4.1 Žb.. Let n s 0, 1, 2, . . . , ␣ g Ž0, ⬁. and let mU2 nq1, 1 be as described abo¨ e. For s 0, Ž1.1. ᎐ Ž1.3. ha¨ e exactly two nonnegati¨ e solutions u 2 nq1, i , i s 1, 2 with 2 n q 1 interior critical points at s 2 nq 1, 1 Ž0, mU2 nq 1, 1 . and s 2 nq 1, 2 Ž0, 0 ., respecti¨ ely. Here 2 nq1, 1Ž0, mU2 nq1, 1 . is gi¨ en by
'
s
2n q 1
'2
ds
H0 'y F Ž s .
q
1
ds
'2 Hmr␣ 'y F Ž s .
or
'
s
m
'2 'y F Ž mr␣ .
.
The corresponding solution is gi¨ en by ds
HuuŽ0.x 'y F Ž s . Ž .
s '2 x ;
x g Ž 0, x 1 .
and ds
HuuŽ x1. 'y F Ž s . Ž .
s y'2 Ž 1 y x . ;
x g Ž Ž 2 n q 1. x1 , 1.
where x 1 s x 1Ž mU2 nq1, 1 . is gi¨ en by x1 s
H0 dsr
Ž 2 n q 1 . H0
ž 'y F Ž s . / dsr ž 'y F Ž s . / y H
m r␣
dsr
ds
ž 'y F Ž s . /
Also 2 nq1, 2 Ž0, 0. is gi¨ en by
'
2 nq1, 2
Ž 0, 0 . s
2 Ž n q 1.
'2
H0 'y F Ž s .
.
123
POSITIVE SOLUTIONS
and the corresponding solution is defined by ds
H0u x 'y F Ž s . Ž .
s '2 x ;
x g w 0, x 1 x
where x 1 s 2Ž n 1q 1 . . Further note that 2 nq1, 1Ž0, mU2 nq1, 1 . - 2 nq1, 2 Ž0, 0. 2Ž nq1.q1, 1Ž0, mU2Ž nq1.q1, 1 .. Remark. Here we note that
2 n , 2 Ž , 0 . - 2 nq1, 1 Ž 0, mU2 nq1, 1 . 2 nq1, 1 Ž 0,
mU2 nq1, 1
. - 2Ž nq1., 1 Ž ,
and
mU2Ž nq1., 1
..
3.5. Ž , . Bifurcation Diagram In this section, given ) 0 and ␣ g Ž0, ⬁., we summarize our results on the existence of nonnegative solution for the semipositone case via the Ž , . bifurcation diagram ŽFig. 3.12. and Theorem 3.5.1. THEOREM 3.5.1. ŽA. Gi¨ en g Ž0, 0, 2 Ž , 0.. and ␣ g Ž0, ⬁., Ž1.1. ᎐ Ž1.3. ha¨ e at least one nonnegati¨ e solution with no interior critical points. ŽB. Gi¨ en n s 0, 1, 2, . . . , g Ž 2 n, 1Ž , mU2 n, 1 ., 2 n, 2 Ž , mU2 n, 2 .. and ␣ g Ž0, ⬁., Ž1.1. ᎐ Ž1.3. ha¨ e at least one nonnegati¨ e solution with 2 n interior critical points.
FIGURE 3.12
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ANURADHA ET AL.
Gi¨ en n s 0, 1, 2, . . . , g Ž 2 nq1, 1Ž , mU2 nq1, 1 ., 2 nq1, 2 Ž , and ␣ g Ž0, ⬁., Ž1.1. ᎐ Ž1.3. ha¨ e at least one nonnegati¨ e solution with 2 n q 1 interior critical points. ŽC.
mU2 nq1, 2 ..
REFERENCES 1. V. Anuradha, C. Brown and R. Shivaji, Explosive nonnegative solutions to two point boundary value problems, Nonlinear Anal. 26, No. 3 Ž1996., 613᎐630. 2. V. Anuradha and R. Shivaji, A quadrature method for classes of multi-parameter two point boundary value problems, Appl. Anal. 54 Ž1994., 263᎐281. 3. V. Anuradha and R. Shivaji, Sign changing solutions for a class of superlinear multiparameter semipositone problems, Nonlinear Anal. 24, No. 11 Ž1995., 1581᎐1596. 4. A. Castro and R. Shivaji, Nonnegative solutions for a class of non-positone problems, Proc. Roy. Soc. Edinburgh, Sect. A 108 Ž1988., 291᎐302. 5. T. Laetsch, The number of solutions of a nonlinear two point boundary problems, Indiana Uni¨ . Math. J. 20 Ž1970r1971., 1᎐13. 6. A. R. Miciano and R. Shivaji, Multiple positive solutions for a class of semipositone Neumann two point boundary value problems, J. Math. Anal. Appl. 178 Ž1993., 102᎐115.