An existence result of positive solutions for singular superlinear boundary value problems and its applications

AN EXISTENCE RESULT OF POSITIVE SOLUTIONS FOR SINGULAR SUPERLINEAR BOUNDARY VALUE PROBLEMS AND ITS APPLICATIONS

YON6-HoONLEE ..

1.

Introduction

In this paper, we are concerned with the existence of positive solu- tions for the boundary value problems of the form;

(1) u"(t) + q(t)g(u(t»

0, 0 < t < 1 u(O)

0

u(l),

where

is singular at 0 and/or

Bya positive solution of (1), we understand a solution u

C[O, 1] n C

(0,1) (or C

(0, 1] or C

[0, 1), depending of the singularity) which is positive on (0,1) and satisfies both the equation on (0,1) and the boundary condition in (1).

Introduce the notation go = tim g(u)

u->o+ U

_ l' g(u) g= - lm - - ,

u->= U

then go = ^{0 and g=} =

correspond to the superlinear problem, and in this case, we show that (1) has a positive solution. This result is motivated by Erbe and Wang [2] who considered regular. problems.

Received January 17, 1997.

1991 AMS Subject Classification: 34B15, 35J25.

Key words: Existence, positive solution, positive radial solution, superlinear problems, singular boundary value problems, generalized Emden-Fowler problems, semilinear elliptic problems, fixed point index arguments.

The Present Studies were Supported by the Basic Science Research Institute Program, Ministry of Education, 1996, Project No. BSRI-96-1410.

(2)

As applications, we first consider Dirichlet boundary value problems of the generalized Emden-Fowler equations

u"(t) +q(t)u(t)P

0, u(o) = 0= u(1),

where pER. When

is not continuous at the end points of (0,1) or

< 0, the problem is singular.

Taliaferro [6] has studied the existence and uniqueness of positive solutions of (2) for

< ° with the shooting method. Zhang [7] has considered the problem for ° ^<

^<

Precisely, under the assumptions

G(O, 1),

> ° ^{on (0,1),} he proved that a necessary and sufficient condition for the existence of positive solutions is

1

s(1 - s)q(s)ds <

using the method of upper and lower solutions.

For the remaining case

> 1, so called superlinear problem, related results have not been founded yet and in this paper, we prove that a sufficient condition for the existence of positive solutions is the same as given in Zhang. We also give a uniqueness result of this problem.

As the second application, we consider the existence of positive ra- dial solutions for the semilinear elliptic problems, in particular, the existence of decaying positive radial solutions with zero boundary con- dition. So we consider

(3) 6u + Ixl->' f(lxl)g(u(x))

0, in n,

u = ^{0, if} Ixl = ^r

^r

> 0, u

as Ixl

where n

{x

Rn : Ixl > ro} and n 2: 3.

Noussair and Swanson [4] obtained positive solutions for the problem of joint dependence nonlinear term, say f (x, u) which is assumed to have polynomial growth in u as u

with bounded continuous coefficients. Bandle and Marcus [1] considered the problems when f ==

1. Noussair, Swanson and Jianfu [5] studied the existence of positive solutions when the nonlinearity involves a critical growth.

In this paper, under suitable conditions on f and

we obtain an

existence result of positive radial solutions for (3) when .A < 2(n - 1).

2. Main results

Let us consider the problem (1). Our main existence result is THEOREM 1. Assume tbe following conditions:

(a1) q E G((O, 1), (0,00» satisfies 10 ¹ s(l - s)q(s)ds < 00.

CbI) yE 0([0;00); [0, M)} sa:tisfies!1o == o and

Tben (1) bas at least one positive solution.

The following lemma is well known and crucial in our arguments, see Guo and Lakshmikantham [3] for proof and further discussion of the fixed point index.

LEMMA 1. Let E be a Banacb space, and let K c E be a cone in E. Assume that 0

and O

are bounded open subsets in E witb °

⁰

and 0

C02. Let T : K n (02 \ (

K be a completely continuous operator such tbat eitber

(i) 11 Tu 11 :S lIull, u

K n 80

and IITull

lIull, u

K n 80

or

(ii) IITull

lIull, u

K n 80

and IITull :S lIull, u

K n 80

Tben T bas a fixed point in K n (0

Proof of Theorem 1. First, it is well known that the problem (1) is equivalent to the integral equation

u(t)

1

G(t,s)q(s)g(u(s»ds,

where G(t, s) is the Green's function corresponding to the linear ho- mogeneous problem explicitly written by

( ) {

s(l - t) for 0 :S s :S t G t,s

t(l - s) for t :S s :S

Thus (1) is equivalent to the fixed point equation u=Tu

in E = G([O, 1]), where T : E

E is given by

Tu(t) = 1

G(t, s)q(s)g(u(s»ds.

By the condition (al), T is completely continuous on the cone of non- negative functions in E. We define a cone K in E by

K

{u

Elu(t) ~ ^{0, t}

[0,1], mina _tE[4'4] u(t) ~ ~llulloo},

then it is not hard to check T(K) c K.

Second, by (aI), we may choose 'fJ > 0 so that 'fJ IoI s(l- s)q(s)ds ::; 1.

Since go

0, there exists RI > 0 such that g(u) ::; 'fJU, for 0 < u ::; RI.

Let n

^{= {u}

^{E :} lIulloo < RI}, then n

is bounded open in E and

o ^{E n}

Moreover, let u E K n ^{anI, then u E K,} lIulloo

^{RI, and thus}

Tu(t)

1

G(t, s)q(s)g(u(s))ds

::;1

s(l- s)q(s)g(u(s))ds ::; 'fJ 1

s(l- s)q(s)u(s)ds

::; 'fJ 1

s(l- s)q(s) lIu

oods ::; lIulloo -

Therefore

IITulloo ::; lIulloo, ^for

u

K n anI-

Third, choose fL > 0 such that 1 Il;44 G(~, ^s)ds > 1. Since goo

00,

there exists R > 0 such that qog(u)

flU, for all u

R, where qo

mintE[I/4,3/4j q(t). Let R

max{2RI,4R} and n

{u

E :

lIulloo < R

then n 2 is bounded open in E and 0 ₁ C n2. We show

IITuJloo

lIulloo, ^{for all u}

^Kn n

^{so let u}

^{K and} lIulloo ⁼ R2, then mintE!I/4,3/4j u(t)

1/41\ulloo

R. Thus q(t)g(u(t))

fLU(t), for all t

[1/4,3/4] and

1 (I 1

TU(2)

Jo G(2,s)q(s)g(u(s))ds

2

1

1

~ i

G(2' s)q(s)g(u(s))ds ~ ^fL i

G(2' s)u(s)ds

4 4

fL {4

1

~ 4: ^J

^{G(2' s)lIu}

^ds > lIulloo.

4

Therefore IITul\oo

Ilul\oo, for u E Knan

and by Lemma 1, T has a fixed point u in Kn(02 \(h) such that RI :::; I\ulloo :::; ^R2. Futhermore, since G(t, s)q(s) > 0 for all s

(0,1), it follows that u > 0 on (0,1), and this completes the proof.

Let

> 1 and let g(u)

up. Theng. obviously satisfies (bd in 'Theorem r ahd we obtaiIl! an~xlsierice resUlt of 'positive solutions for "

(2) as follows;

COROLLARY 1.

Let p > 1 and assume (al) in Theorem 1.

Then (2) has at least one positive solution.

We have a similar result as Theorem 1 when

is singular at O.

THEOREM

2. Assume (b

in Theorem 1 and (a2) q

G((O, 1], (0, 00)) satisfies Io ¹ sq(s)ds < 00.

Then (1) has at least one positive solution.

Proof. The proof generally follows that of Theorem 1, and it is enough to check the second part in the proof of Theorem 1.

By (a2), we may choose TJ > 0 so that TJ I; sq(s)ds :::; 1. Since go = 0, there exists RI > 0 such that g(u) :::; TJU, for 0 < u :::; RI. Let 0.

{u

E: I\ul\oo < RI}, and let u

K n anI, then

Tu(t) = ^{(1- t)} 1

sq(s)g(u(s))ds + t 1

^(1- s)q(s)g(u(s))ds :::; (1- t) 1

sq(s)g(u(s))ds

:::; TJ 1

sq(s)u(s)ds

:::; TJ 1

sq(s)l\ul\oods :::; I\ul\oo.

Therefore

I\Tul\oo :::; I\ul\oo, ^{for all u E K} n anI,

and the proof is done.

Since the problem having the singularity at 0 and the problem hav-

ing the singularity at 1 are equivalently transformed, we get the fol-

lowing corollary;

COROLLARY

2. Assume (br) in Theorem 1 and (a3) q E C([O, 1), (0, 00» satisfies Jo 1

(1- s)q(s)ds < 00.

Then (1) has at least one positive solution.

3. Uniqueness

Let u be a positive solution of (1) and let L

= maxtE[O,lj g(u(t».

Then q(t)g(u(t» ::; Lnq(t) and J~ lu"(t)ldt ::; Ln J~ q(t)dt < 00, pro- vided by

L

[0, 1]. Thus both u'(O+) and u'(l-) exist and conse- quently, all positive solutions of (l)are of Cl [0,1] nC 2(0, 1). Based on this fact, we obtain the existence of a unique positive solution for (1) as follows;

THEOREM

3. Assume

(a4) q

C«O, 1}, (0,00» satisfies J~ q(s)ds < 00.

(b

9 E C([O, 00), [0,00» satisfies

= ° ^{and goo = 00.}

(b

9 is increasing and g~u) is strictly monotone.

Then (1) has a unique positive solution.

Proof Suppose that (1) has two distinct positive solutions U1 and U2. We first show that there exist two ordered positive solutions of (1). Let <j>(t) = min{u1(t),U2(t)}, t

[0,1]. Using the fact that

is increasing, G(t,s),q(t) 2: 0, we get

T<j>(t) = 1

G(t; s)q(s)g(<j>(s»ds

::; min{ r ¹ G(t; s)q(s)g(ul(s»ds, .11 G(t, s)q(s)g(u2(s»ds,}

Jo - Jo .

= ^{min{U1(t), U2(t)}} = <j>(t)

Thus

decreases to a positive solution U3 of (1) and U3(t) ::; min{u1(t), U2(t)}, for all t

[0,1].

Therefore we have two ordered positive solutions U3 and one of U1 or

U2 whether U1 and U2 are ordered or not. So for convinience, assume

that u and v are two ordered positive solutions of (1) with u(t) ::; vet),

(3)

for all

[0,1]. Let z(t) = u(t)v'(t) - u'(t)v(t). Since u, v

G

[0, 1], z(O)

0

z(l) and

z'(t)

u(t)v"(t) - u"(t)v(t)

=,

q(t)u,(t)v(t)(g~~~i) ^- g~~(~;))) ^{on (0,1).}

Since f';:) is strictly monotone, either

z'(t) > 0 or z'(t) < 0 on (0,1).

This contradicts to z(O)

0

z(l), and the proof is complete.

Finally, we obtain the uniqueness of positive solutions for the gen- eralized Emden-Fowler equations.

COROLLARY

3. Let p > 1 and assume (a4) in Theorem 2.

Then (2) has a unique positive solution.

4. Existence of positive radial solutions

Let us consider the semilinear elliptic problems of the form;

/::,.u + \xl-Af(lxl)g(u(x)) = 0, in n,

u = 0, if Ixl = ^ro,

u

as Ixl

00,

where n

^{x

^{Rn :} Ixl > ro} and n

3. For any real number A satisfying A < 2(n - 1), we prove the existence of positive radial solutions for (3) if

and f satisfies the following conditions;

(bd

E G([O, 00), [0,00)) satisfies go

0 and goo

00.

(cd f

G([r o, 00), (0, 00)) satisfies Ire: ^x

^{f(x)dx < 00.}

We are concerned with radial solutions, thus for the radial variable

r

= lxi, we write (3) as

n - l .

(3') u"(r) + --u'(r) + f(r)g(u(r)) = 0,

r

u(r

0,

u(r)

0 as r

00.

Setting s

r2- n, v(s)

u(r(s)), and then

z(t) = v(s), we rewrite (3') as

z"(t) + q(t)g(z(t))

0, z(o) = 0= z(l),

r2->. -2(n-1)+>' -1

where q(t) = (n--2)2 (1- t)

f(r

(1- t)

Thus by (Cl), q

C([O, 1), (0, (0)) is singular at 1 satisfying 1;(1 - s)q(s)ds <

and for the problem in this section, it suffices to consider the existence of positive solutions of the problem (3) with the conditions on

described above. Therefore by Corollary 2, ,we obtain an existence result for problem (3).

COROLLARY

4. Let A < 2(n - 1), and assume (b l ) ^and (Cl).

Then (3) has at least one positive radial solution for all 0 < r

<

It is easy to check that if Ir": xn-l-Af(x)dx <

then 1 ₀ ^{1 q(s)ds} <

00.

Thus we obtain a uniqueness result for (3) as follows;

COROLLARY

5. Let A < 2(n - 1), and assume (b

and (b

More- over, assume

(C2) f

C([r o, ^{(0), (0, (0)) satisfies} Ir": xn-l-Af(x)dx <

Then (3) has a unique positive radial solution for all 0<

<

EXAMPLE.

If g(u)

uP, p> 1 or g(u)

eU-u, then (3) has at least one positive radial solution or a unique positive radial solution provided

fooXl-A

f(x)dx <

or

xn- l - Af(x)dx <

respectively.

Jro Jro

References

1. Bandle C. and Marcus M., The positive radial solutions of a class of semilinear elliptic equations, J. reine angew. Math. 401 (1989), 25-59.

2. Erbe L. H. and Wang H., On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc. 120(3) (1994),743-748.

3. Guo D. and Lakshmikantham V., Nonlinear Problems in Abstract Cones, Aca- demic Press, Orlando, FL, 1988.

4. Noussair E. S. and Swanson C. A., A positive solutions of semilinear elliptic problems in unbounded domains, J.Diff. Eqns. 57 (1985), 349-372.

5. Noussair E. S., Swanson C. A. and Jianfu Y., Positive finite energy critical solutions of semilinear elliptic problems, Can. J. Math.44 (1992), 1014-1029.

6. Taliaferro S. D., A nonlinear singular boundary value problem, Nonlinear Anal- ysis, T.M.A. 3 (1979),897-904.

7. Zhang Y., Positive solutions of singular sublinear Emden-Fowler boundary value problems,J. Math. Anal. Appl. 185 (1994), 215-222.

Department of Mathematics .Pusan Nationaf"dnifersltY .

Pusan 609-735, Korea

E-mail: [email protected]