EXISTENCE AND MULTIPLICITY OF POSITIVE RADIAL SOLUTIONS FOR SEMILINEAR ELLIPTIC-EQUATIONS IN ANNULAR DOMAINS

(1)

SIAM J. MATH. ANAL.

Vol. 22,No. 6,pp. 1500-1515, November 1991

1991Society for Industrial and Applied Mathematics 002

EXISTENCE AND

MULTIPLICITY OF POSITIVE

RADIAL SOLUTIONS

FOR SEMILINEAR ELLIPTIC

EQUATIONS

IN

ANNULAR

DOMAINS*

SONG-SUN LIN’$ AND FENG-MING PAIr

Abstract. The existence and multiplicity of positive radial solutions of equationAu+f(u) 0is studied inannular domains inR

n,

n>_- 2.Itisproven that iff(0)>_-_0,fissomewherenegativein(0, )and superlinear at

,

then thereis alargepositive radial solutionsonall annuli.Iff(0)<0and satisfies certainconditions, then the equation has no solution if the annuliare toowide. Multiplicityresults are also obtained when_f has manyhumpswith positive areas.

Keywords, elliptic, semilinear, positiveradial solution, annular domain

AMS(MOS)subjectclassifications. 35B32, 35JB65,35P30

1. Introduction.

In

thispaperweconsidertheexistence andmultiplicityofpositive

radial solutions ofthesemilinear elliptic equation

(1.1)

Au(x)+f(u(x))=O

ina<]xl<b,

(1.2)

u(x):O

_onlxl=a

and

Ix[:b,

x

",

n_->2and

_{f C((0,}

c))fq

C([0,

))

satisfying the following hypotheses"

(HI)

fis

negative somewherein

(0, );

(H2)

fis

superlinear at u

=,

i.e.,

limu_f(u)/u

=.

One of the problems forsemilinear elliptic equations in annular domains which have been studied quite extensively in recentyears is"

(P)

Does

(1.1),

(1.2)

possessa positive radial solution inevery annulus?

The answer to

(P)

was proved affirmative by Nehari

[20],

assuming that

_f

is

positive in

(0, c)

and satisfies the condition" ::i6>0 suchthat

f(u)/u

+ ismonotone

increasing in

(0,

).

Later, (P)

was studied by Kazdan and

Warner [15],

Ni and Nussbaum

[21],

Bandle,

Coffman, and

Marcus [2],

Garaizar

[13],

and Lin

[17].

In [2], Bandle,

Cotiman,and

Marcus

showed thattheanswer to

(P)

isaffirmative, provided that

fis

positive in

(0,

c)

and satisfiesthe following conditions"

(A1)

fis

nondecreasing in

(0,

);

(A2)

lim,__,of(u)/u

=0;

(A3)

lim,_.f(u)/u

=.

In [2],

it is remarked that

(A1)

is not a necessary condition for existence. This have been confirmedby Coffman and

Marcus [8]

and Lin

[17]

independently.

With a suitablechange of independent variable,

(1.1),

(1.2)

become equationsof

the form

(1.3)

u"(t)+G(t,u)=O,

to<t<tl,

(1.4)

u(to)

0

u(tl).

* Received by the editorsJanuary22,1990; acceptedfor publicationJanuary 7, 1991.

?DepartmentofAppliedMathematics,National ChiaoTungUniversity,Hsin-chu,Taiwan, Republic of China.

The work of this author was partiallysupported bythe National Science Council of the Republic of China.

1500

(2)

In

[3],

Bandle and

Kwong

showed that the answer to

(P)

is affirmative ifG satisfies the followingconditions:

(G1)

G is CO

in its first variable and C in its second;

(G2)

limu_o

G(t, U)/U

oO _uniformlyon every to,

tl];

(G3)

limu_o

G(t, u)/u

=0 uniformlyon every

[to, tl].

G(t, u)

is now allowed to be negative for small positivevalue and

G(t, 0)=0

is assumed implicitlywiththe limit involved exists andfinite in

(G3).

In

this paper, we first generalize the results ofBandle and

Kwong [3],

showing that

(P)

is affirmative if

(H1),

(H2),

and

(H3)

are satisfied.

(U3)

f(0)_->

0.

Moreover,

solutions obtained are"large" inthe followingsense:

By

(H1),

thereexists

(u.,

u*)c

(0,

oo)

such that

(1.5)

f(u)>-O

in

(u*,

oo),

f(u)<0

in

_(u.,

u*),

f(u.)=f(u*)=O.

Let _3’

> u*

bethe smallest number such that

(1.6)

f(u)

du =0.

u,

The solution u of

(1.1),

(1.2)

is calledlarge if

(1.7)

_Ilull--

max

{u(x):

a<-Ixl<-b}

>

.

On the contrary, Garaizar

[13]

showed that

(P)

is negative, i.e.,

(1.1), (1.2)

has no positive radial solution if b-a is too large, if

_f

satisfies the following conditions:

(i)

f(0) <

0;

(ii)

There exists t>0 such that

F(u)<=O

in

(0, t)

and

f(u)>0

in

(tT, );

(iii)

There exists k>l and

d2->dl>0

such that

dluk<=f(u)<-d2u

k for u large,

where

F(u)=

f(t) at.

We

can alsoobtain a similar nonexistenceresult withoutassumingcondition

(iii),

i.e., if

(H2)

and thefollowing hold true:

(H3)’(i)

f(0)

<

0;

(ii)

There exists t

>

0such that

F(u) <

0 in

(0, tT]

and

f(u) >

0 in

(tT,

c).

On

the other

hand,

when

_f

changes signs, the existence of multiple positive

solutions of the equation

(1.8)

Au+Af(u)=O

inf,,

(1.9)

u 0 on

h_-->0 and fl is a bounded smooth domain in

n,

n_->2, has been studied by many authors

(see,

e.g.,

Brown

and Budin

[5], Hess [14],

de Figueiredo

[12],

Clement and

Sweers

[6],

and

Wang

and Kazarinott

[24]).

In [14], Hess

showedthat if

_f

satisfies the followingconditions:

(fl) f(0) >

0;

(f2)

There exist m numbers a,,>t,,_l>...>t71>0 such that

f(ak)=O

for k= 1,"

,

m,

(f3)

O<max{F(s):O<=s<=ak-}<F(ak),k=2,’’’,

m,

thenthere existsanumber

>

0such that for allA

>

,

(1.7),

(1.8)

have atleast 2m 1 positive solutions

if1,

u2,

a2,

Urn,

am

such that

_Ilalll

</1

and/k-1 <

Ilull,

Ilall

<

_a

for

k=2,...,

tn, and

k-l(k

and Uk(

k

for

k=2,...,

m.

(3)

1502 SONG-SUN LIN AND FENG-MING PAI

Later,

de Figueiredo

[12]

obtainedthe existence of 2m-1 ordered positive sol-utions under slightlydifferentassumptions.

In

thispaper,weshowthat

_iff

satisfies

(f2)

and

(f3),

thenthereexists

b,, >

a such

thatfor anyb

> b,,,

(1.1),

(1.2)

has atleast rn ordered positiveradial solutions Ukwith

Ilu ll

)

for

k=2,...

,m.

Moreover,

if

f(0)>=0,

then

(1.1), (1.2)

has atleast 2m- 1 positive radial solutions for b

>

For

theother related problems, notethe following:

(i)

Uniqueness of positiveradialsolution, when

f(u)

>

0for u

(0,

c),

has been studied by Ni and Nussbaum

[21], Bandle,

Cottman,

and

Marcus [2],

Bandle and

Kwong

[3],

andCottman and

Marcus

[8].

(ii) Symmetry

breaking for positive radial solutions has been studied by Brezis and Nirenberg

[4],

Coffman

[7],

Suzuki and Nagasaki

[22], [23],

and Lin

[16], [18],

[19].

The methods usedin thispaperareshooting techniques, thephase-plane

method,

and variational methods.All results obtained in this papercan also begeneralizedto

f(r,

u)

which satisfies certainuniformity assumptions in r as in

(G2)

and

(G3).

The paper is organized as follows.

In

2, we obtain some preliminary results which areuseful.

In

3,weprovethat

(P)

isaffirmativewhen

(H1)

(H3)

aresatisfied.

In

4, we prove

(P)

is negative when

(H1)--- (H3)’

are satisfied.

In

5,

we obtainthe multiplicity results forwide annuli.

2. Preliminaries. Since we are interested inpositive radial solutions of

(1.1),

we write

(1.1),

(1.2)

in the form

n-1

(2.1)

u"(r)+ u’(r)+f(u(r))=0

in

(a,

b),

(2.2)

u(a)=O=u(b).

For

fixed a

>

0, we consider the family ofsolutions

u(. ) u(., a)

of the initial

value problem

n-1

(2.3)

u"(r)+

u’(r)+f(u(r))

=0 for

r>

a,

(2.4)

u(a)=O

and

u’(a)=

where a=>0 is the shooting parameter.

Furthermore,

(2.3), (2.4)

can also be written asa dynamical system

(2.5)

u’=v,

n-1

(2.6)

v’=

v

-f(u),

with initial data

(2.7)

u(a)=O

and

v(a)-

a.

We

define an energyfunction

H(. )= H(u(., a))

by

(2.8)

H(r)

1/2v2(r) +

F(u(r)).

Then,

along each trajectory of solution of

(2.5), (2.6),

H isdecreasing; in

fact,

n-1

(2.9)

H’(r)

u’2(r)

_-<0.

Furthermore,

H(r)

is strictly decreasingin r if a

>

0.

(4)

We first classifythe solutions

u(.,

a).

DEFINITION 2.1.

For

any a_-->0, a belongs to one ofthefollowing three disjoint

sets:

(i)

a

P

(or u(., a)

is a

P-solution)

if

u(r, a)>

0for all r

>

a,

(ii)

a

N

(or

u(.,

a)

isan

N-solution)

ifthere exists

b(a)

>

0such that

u(r, a)

>

0 in

(a,

b(a)), u(b(a), a)=0

and

u’(b(a), a)<0,

(iii)

a

T

(or u(., a)

isa

T-solution)

if there exists

b(a)

>

0such that

u(r, a) >

0 in

(a, b(a)),

u(b(a),

a)=0

and

u’(b(a), a)=0.

We

thenstate some simple butbasicproperties ofsolutions

u(., a).

LEMMA

2.2.

(i)

If

O

N,

then

u(., a)

has

only

onelocalmaximum.

(ii)

If

aSCJ T,

then

H(u(r,

a))>O forr(a,b(a)).

(iii)

If

(H2)

is

_satisfied

andu r,a

>

0

_for

allr

>

_ro

>--

a, thenu r,a is bounded.

(iv)

Nisan openset.

Proof.

(i)

The proofof

(i),

inthe general case,was given byGaraizar

[13].

The main idea isusing energy

H(r),

which decreasesalong the trajectory,andthenobtain

the following two facts:

(a)

the trajectory cannot cross

(intersect)

itself;

(b)

the trajectorycannotbe tangent to the u-axis.

Therefore, (i)

can beproved.

For

the details, see

[13, Lemma 1].

(ii)

Since

H(u(b(a),

a))->0,

(ii)

follows.

(iii)

Since

H(u(r,

a))-1/2u’Z(r)/

F(u(r, a))- H(u(a, a))=

az/2,

we have

F(u(r,

a))-az/2

for all

r-ro.

Therefore,

(H2)

implies

u(r, )

is bounded.

(iv) By

the Implicit Function

Theorem,

b(a)

is continuouslyditterentiable in N

and Nis an open set.

The following lemma indicates there is agreat difference between cases

f(0)_->

0 and

f(0)

<

0.

LEMMA

2.3.

_If

f(O)

>--

_O,

then

T

qb.

Furthermore,

if

a

T

then

u(r,

a)

>

0

_for

all

r>b(a).

Proof.

If

f(0)=0,

then

(u,

v) =(0,0)

is anequilibrium.

Hence,

T=b.

If

f(0)>0

andtherewerea

T,

then

u"(b(a),

a)

-f(0)

<

0.

Therefore,

u(r,

a)

<

0forr

< b(a)

andsufficiently closeto it, a contradiction. This proves

T-b.

If a

T,

then it is necessary that

f(0)

<0. Since

H(u(b(a),

a))=0

implies that

H(u(r, a))<0

for all

r> b(a),

then

u"(b(a), a)= -f(0) >

0 implies that

u(r,

a)>0

for all

r>

b(a).

[2

The followinglemmaplays acrucial role inthe study of problem

(P).

LEMMA

2.4.

_If

thereis a sequence

{ak}

C N(3

T

such that

ak>O

and

b(ak)-o

as

k->,

then 6

P

and

u(.,

6)

satisfies

the followingmonotonicityproperty:

(M)

(i)

u(r,

6)

is eitherstrictly increasing in

(a, c)

or there exists al

>

a such that

u(r,

6)

isstrictly increasing in

(a,

a

l)

andstrictly decreasing in

(al,

oo).

(ii)

u(

r,6

-

as r-->oo where

f( 6)

O.

Proof.

First, we observe that

u(., 6)

cannot have a local maximum followed by

a local minimum. Otherwise, by continuous dependence of ordinary differential equations

(o.d.e.),

for k sufficiently large,

u(r, ak)

will have at least two local maxima in

(a, b(cck)),

a contradiction to Lemma

2.2(i). It

is also clear that

u(r,

6)

cannotbe constant on any finite interval of

(a, oo). Hence, u(., 6)

satisfies

(M)(i).

Condition

(M)(ii)

follows by Lemma2.6 which will be provedlater.

(5)

As

in

[2], [3],

and

17],

it is sometimes convenient tostudy theexistenceproblem

inthe form of

(1.3),

(1.4).

For

n_->3, in terms of variables

(2.10)

s r2-n and

w(s)

u(r),

equations

(2.1), (2.2)

can be rewritten as

(2.11)

w"(s)+p(s)f(w(s))=O

in

(So, S1),

(2.12)

W(So)

=0=

w(s,),

where

p(s)=(n-2)-2s

-,

k=(2n-2)/(n-2),

So b

-",

and

s=

a

-".

For n=2, in terms ofvariables

s=1/2-1oga+logr

and

w(s)=u(r),

equation

(2.1)

can also be written as

(2.11)

with

p(s)=a

e

2s-, So=1/2

and

s=

loga+log b. In the remaining part ofthe section, we only treatthe case n>_3;

2

the case n 2 can also betreated analogously.

The associated initial valueproblem, nowbackward shooting in ans-variable, is

(2.13)

w"(s)+p(s)f(w(s))=O

fors

<

s,,

(2.14)

w(sl)=0

and

w’(sl)=-/3,

a2-" is afixed number.

where/3

>_{0 is} _{the shooting parameter} _and

_s

It is easy to check that

(2.13),

(2.14)

isequivalentto

(2.15)

w(s)=fl(Sl-S)-

(t-s)p(t)f(w(t))dt

fors

<

Sl,

and the solution

w(.,/3)

also satisfies thefollowing equation"

(2.16)

w(s)=w(g)+w’(g)(s-g)+

(t-s)p(t)f(w(t)) at

forO<s,g<Sl.

The associated energyfunction V isdefinedby

V(s)=-

V(w(s,

))=1/2w’(s)+p(s)F(w(s)).

(2.17)

It is clear that and so

V’(s)=p’(s)F(w(s)),

(2.18)

V(s)

V(g)

+

p’(t)F(w(t))

dt for 0

<

s,g

_<

_s.

Ifw has a zero in

(0, s),

denote

So(fl)=inf{so:

w(s,/3)

>

0 in

(So,

sl)},

and

,(/3)

(So(/3), sl)

satisfies

w( v(/3 ),

/3

max

{w(s,/3):

s

(So(/3), s)}.

With a modification of the argument used in Lin

[17],

we can prove that

So(fl)

and

,(/3)

are well defined for sufficiently large

/3

and tend to

_s

as

/3

c. For completeness, we also give a fullproofhere.

(6)

LEMMA2.5.

_If

condition

(H2)

issatisfied, then

So(

and

v(

arewell

_defined

_for

sufficiently large

ft.

Moreover,

(2.19)

lim

p(/3)

(2.20)

lim

So(fl

s

and

(2.21)

lim

w(v(), fl)=oo.

Proof.

We

first prove

(2.19).

If

(2.19)

were

false,

then there would be a point

Vo

(0,

Sl)

and a sequencefig c with

(2.22)

Wk(S)

>

0 and

W’k(S)

<----

0 in

(Vo,

sl),

where

Wk(S)= W(S,

k).

Letting g=

(Uo+

s)/2,

we claimthat

(2.23)

limsup Wk(g) o0.

k--->

Suppose

this is notthe case; then there exists a constant M>0 such that

(2.24)

Wk(g)<-

m

forall k.

Now,

by

(2.16)

and

(2.24),

we have

Wk(g)=-flk(Sl--VO)

(t--g)o(t)f(wk(t))

_{dt>--flk(Sl-Vo)-C,}

forsome constant C_-> 0.

But,

by

(2.24),

this is impossible.

Therefore,

(2.23)

holds.

By

choosing asubsequence

of/k

ifnecessary,we mayassume that

(2.25)

lim Wk

Denote

in

(Vo,

g) and

By

(2.25)

and

(H2),

hk(S) :f(

Wk

(S))/Wk(S)

mk inf

{

hk

(s)"

s Vo,

g]}.

(2.26)

lim mk koo

Now

W"k(S)

+

p(s)hk(s)w(s)

0 in

(Vo,

g)with

p(S)hk(S)

>--

p(g)mk

in

(Vo,

g). By

(2.26)

and the

Sturm

Comparison

Theorem,

Wk has zeros in

(Vo, )

for sufficiently large k.

But

by

(2.22)

this is impossible. Thisproves

(2.19).

Next,

weprove

(2.21).

By

(2.18),

we have

1_ [32k

p(Vk)F(U(Vk))+

p’(t)F(Wk(t)) dt,

2 _k

where Vk

V(flk),

whichimplies that

F(Wk(Pk))aZ

as kc.

By

(H2), (2.21)

follows.

(7)

1506 SONG-SUN LIN AND FENG-MING PAl

Finally, weprove

(2.20).

If

(2.20)

were

false,

thenthere would beapointSo6

(0, sl)

and asequence

/3k

with

(2.27)

Wk(S)

>

0 in

(So,

’k)-Denote

g=1/2(So+

sl). By

(2.19),

wemay assumethat g<_Vk_for_all_k._We_first_claim_that

(2.28)

lim sup Wk

(g)

< o.

Let

Lk=min {wk(s)"

s [g,

Then,

there exists L>0 such that

(2.29)

Lk

=<

L for all k.

Otherwise, by the Sturm Comparison Theorem again, wk has a zero in (g,

uk),

a

contradiction to

(2.27).

Ifwk(g) Lk,then

(2.28)

holds.

If Wk(g

>

Lk,letS_k

(g, lk)

suchthat

Wk(Sk)

_Lk.

Denote

rk

S/(2-n)

=

gl/(2-n) and Uk Wk.Then

u(rk)=O,

and we have

n(uk(rk))

F(Lk)

H(Uk(f))

F(Uk(f)).

By

(U2)

and

(2.29),

Uk()

M

for some constant M

>

0. This proves

(2.28).

By

(H2),

there exists

u*

>

0such that

f(u)

>

0for all u

>

u*.

Denote

Ak

{s

(0,

rk)"

Wk(S)

U*}.

Then by

(2.16),

we have

Wk(Uk)

Wk(g)+ W(g)(Uk--g)+

(t--

k)p(t)f(wk(t))

dt Wk(g)+

W(g)(Pk--g)+

(t--

k)p(t)f(wk(t))

dt dA

w()

+

w(s)(

)

+

c

for some constant C 0.

Hence,

by

(2.20),

we have

(2.30)

lim

w(

g)

.

On the other

hand,

by

(2.16)

again, wehave

(so

(

+

((so-

+

(

soo(f((t

(

-

((s

so

(

soo(f((

a

1

(

-

(s,-

so(

+

Cl

(8)

for some constant C1_->0.

By

(2.28)

and

(2.30), Wk(So)’-’)--(X3

as

k-*o,

acontradiction to

(2.27).

This proves

(2.20).

F1

LEMMA

2.6.

_If

u(r, )>

0

_for

r

>

_ro

>-_a, then

(2.31

lim inf

lu(r,

a

Z

0,

where

Z

{a

>-0:f()

0}

and

_lu-Zl

inf{lu-

al:

a

z}.

In

particular,

_if

limr

u(r, a)

_->0, then

f()

O.

Proof

If

(2.31)

were

false,

then there wouldbe an e

>

0 suchthat

[f(u(r,a))l>-e.

Denote

w(s, fl)= u(r, a).

Then by

(2.15)

Iw( ,

as s0

+.

This is impossible inviewing

w(s,/3)

>

0 and

Lemma 2.2(iii).

l-1

3. Existence of large solutionswhen

f(0)>_-0.

In

thissection we shallprove that

_if

(H 1)

(H3)

aresatisfied, the answerto

(P)

is

_affirmative.

We

_first

prove thefollowing lemma.

LEMMA

3.1.

_If

a S and u

II-->

then

(3.1)

_Ilull

>

),,

where Tis in

(1.6).

A

similarresult holds

_for

a

T

with

(3.2)

max{u(r,a):re[a,b(a)]}>=

y.

Proofi

Ifae

N,

by

Lemma 2.2(i),

there existsaunique

r(a)e (a,

b(a)),

suchthat

(3.3)

u(r(a),

_Ilull.

Let

rl(a)e (r(a), b(a))

such that

u(rl(a),a)=u.,

which implies

’u(())f(u)du

>

O. u,

Hence

u(r(a))>

3’. This proves

(3.1).

By

the same argument, we can obtain

(3.1)

if a

T

and

(3.2)

holds.

LEMMA

3.2.

Assume

conditions

(H1)---(H3)are

satisfied.

Then

(3.4)

N1

{a

e

N:

_Ilu(.,

is a nonempty open set.

Proof.

By Lemmas

2.3 and 2.5,

_N

is nonempty.

By

Lemma 3.1 and continuous

dependenceof

o.d.e.,

N1

is an openset.

We

can nowprove the main result ofthis section.

THEOREM3.3.

Assume

conditions

(HI)--- (H3)

are

_satisfied.

Then

_for

anyb

>

a

> O,

there exists apositive radial solution u

(r)

of

(2.1),

(2.2)

with

Proof.

By

Lemma 3.2, there exists a*_>₀ _{such that}

_N1D

_{(a*, oo)}

_with

_a*e

_N1.

where

_u.

is in

(1.5).

Then

H(u(r(a)))

F(u(’(c)))

>

H(U(rl(a)))

>

F(u.),

(9)

By

Lemma 2.5, it suffices to showthat

(3.5)

lim

b(a)

(*)+

We

shall prove thetheorem accordingto

f(0)>

0 and

f(0)=

0.

If

f(0)>

0, we claimthat

a*>

0.

In fact,,

u"(a, 0)=-f(0) <

0.

Hence,

there is an e>0 suchthat

u(r, 0)

<0 for

r(a,a+e).

Thisimplies

a*>0. We

claimthat

a*P.

Ifa*

P,

then

(0, c)=N U

P

implies

a*

N.

Since

a*

N,

wehave

u(z(a), a)

>_y.

By Lemma

3.1,

a* N1,

acontradiction.

Therefore, a* P

and

(3.5)

follows.

If

f(0)=0,

then either

a*>0

or

a*

=0. If

a*>0,

then the previous argument

also works and then

(3.5)

holds. If

a*=0

and

(3.5)

are

false,

then there are

_bo>

a

and 6

>

0 such that

b(a)

_-<

_bo

for all a

(0,

6).

Since

z(a)

(0,

bo)

for all a

(0, ),

thereexists asequence

akO

such that

Z(ak)-

Zo

[0, bo].

Since

U(’(ak),

ak)>

y, we

have

U(Zo, 0)=>

y, a contradiction to

u(r, 0)-=

0.

Hence, (3.5)

holds.

COrOLLArY 3.4.

Assume

conditions

(HI)

(H3)

are

_satisfied

and

f(O)

>

O. Then

for

any a

> O,

the equation n-1

(3.6)

u"(r)+u’(r)+f(u(r))=O

in

(a, c),

(3.7)

u(a)=0

and

u(r)>0

forr>a,

hasa solution u which

_satisfies

(M).

Proof

In

the proof of the previous

theorem,

we have

N1

=

(a*, c)

with

a*>

0 and

a*

P. By

(3.5)

and

Lemma

2.4,

u(.,

a*)

satisfies

(M).

4. Nonexistence on wide annuli when

f(0)<

0.

In

this section we shall prove that

if

(H2)

and

(H3)’

are satisfied, then

(2.1),

(2.2)

has nopositive solution when b-ais too large.

We

first show that P

₄

when

f(0)

<

0.

LEMMA

4.1.

_If

f(O) <

O,

then there exists

_a.

>

0 such that

[0,

a.)

P. Proof

We

first prove 0eP.

In fact,

u"(a,O)=-f(O)>O

and

H(u(a,O))=O>

H(u(r,O))

for

r>a

implies

u(r, 0)>0

for

r>a. Hence

0eP.

Next,

let Uo

>

0 such that

f(u)

<

0 in

[-Uo, Uo].

Then there exist e

>

0and

ao

>

0 such that for all

a[O,

ao],

we have

lu(r,a)l<-_Uo

in

[a,a+e]. Therefore,

for all

[0, no],

n-1

u"(r)+u’(r)>O

in[a,a+e],

which implies

u(r, a) >

0 in

(a,

a

+

e

].

On

the other

hand,

if

H(u(a

+

e,

0))

<

0, bycontinuousdependence of

o.d.e.,

there

exists

_a.

(0, no)

such that

H(u(a

+

e,

a))

<

0 for all a

(0,

a.).

Therefore,

for any

a(O,a.),

H(u(r,a))<O

for

r>a+e,

which implies

u(r,a)>O

for

r>a+e.

This

proves

(0,

a,)c

P.

[3

We

now prove themain resultofthis section.

THEOREM 4.2.

Assume

conditions

_(H2)

and

(H3)’

are

_satisfied.

Then there exists

b* >

asuch that

_for

any b

> b*, (2.1), (2.2)

has no positive solution.

Proof.

By

Lemma 2.5,there exist

a*>0

and

_bo>

a such that

(a*, )c NU

T

and

b(a)<-bo

for all a

(a*,c).

On the other

hand,

by

Lemma

4.1, there exists

a,>0

such that

[0,

a,)c

P. Therefore,

it suffices to showthat there exists

_bo>

a such that

(4.1)

b(a)<-b’o

forany

a[a.,a*]O(NT).

(10)

If

(4.1)

were

false,

then there would be a sequence Ck

[a.,

c*]

f’)

(NU T)

and

c

Ice,,

ce*]

suchthatCek--> and

b(ak)->

cas k->cx3.

By Lemma

2.4,c6Pand

u(.,

satisfies

(M). Now,

(H3)’(ii)

and

u(r, )-

as r-o

_withf(t)=

0implies

H(u(r,

))

F(t)<0

as

ro. Therefore,

there exists

_ro>

a such that

H(u(ro, c))<0.

By

con-tinuous dependence of

o.d.e.,

we have

H(u(ro, ak))<O

for k sufficiently large, a contradiction to

Lemma 2.2(ii).

This proves

(4.1).

5. Multiplicity results on wide annuli.

In

the previous sections we studied the existence oflarge

(and

nonexistence

of)

positive solutions for

(2.1), (2.2)

underthe various assumptions

_off

In

this section weshall study the existence of"intermediate size" solutions of

(2.1),

(2.2)

when

_f

may

change

signs several times and satisfies condition

(f3)

in 1, i.e.,

f

satisfies thefollowing hypothesis:

(H4)

there exist m successive numbers /m

>

/m--1>"

>

/1

>

0,which satisfy

(i)

f(tTk)=0

for k= 1,..., m; and

(ii)

O<max

{F(s)’O<--s<=ak_}<F(fk)

for

k=2,

m.

Let

yk be the least number in

(_,

t)

such that

(5.1)

f(u)

du =0,

for k 1,..., m, where

ao-=

0.

We

firstprove the followinglemma.

LEMMA

5.1.

Assume

there exists an

>

0 such that

(5.2)

f(u)

=0

_foru

=

.

Then we have the following conditions:

(i)

_if

u(rl,

a)

a

and

u’(rl, a)

>- 0

_for

some

_r

>

a, then

_for

r

>

r,

1 1

(5.3)

u(r,

n-2 n-2

(ii)

let

U

{a

(0, c): u(rl, or)

_{a for}

some rl

>

a};

then there exists

a* >

0 such

that

U

a

*,

).

Proof

(i) By

(5.2),

we have

n-1

(5.4)

u"(r)+u’(r)=O

as long as

u(r, a)

>-

.

_Therefore,

by solving

(5.4)

with initial condition

u(rl,

a)=

t7

and

u’(rl, or)_->0,

(5.3)

follows.

(ii)

We shall prove

(ii)

by using the method of backward shooting. If

(ii)

were

false,

there would be asequence fig-* such that

Wk(S)

<

aslong as Wk remainpositive,

where

Wk(S)= W(S,

ilk). Let

Vk inf

{g

(0, S1): Wk(S >

0and

W’k(S)

<----

0 in

(g,

S1)}.

We

claimthat

(5.6)

lim /.tk s

If

(5.6)

were

false,

there would be

_Uo<S

and a subsequence of Uk

(for

simplicity,

(11)

SONG-SUN LIN AND FENG-MING PAl

renameit

Uk),

suchthat /"k /’0forall k.

Therefore,

by

(2.15)

and

(5.5),

we have

Wk(VO)

flk(Sl-- VO)--

(t--

Vo)p(t)f(wk(t))

’0

>-

_{(s,- o)-}

C,

forsome constant

C->0,

a contradiction to

(5.5).

Hence

(5.6)

holds.

On

the other

hand,

by

(5.5)

again, wehave

W’(rk)

_{--ilk +}

o(t)f(wk(t))

dt

=<-]

+

C

for some C->0.

Therefore,

W’(Zk)<0

if k is large enough, a contradiction to the

definition of Vk. This proves

(ii).

VI

The

(energy)

functional we want to minimize is

J(u)

r

"-

u’(r)-

F(u(r))

dr

in

H((a,

b)),

where

H((a,

b))= {u’u

isabsolutelycontinuous in

[a, b]

with

u(a)=

0=

u(b)

and

u,

u’

L2(a,

b)}.

Since

_f

may change signs, the minimizer _Ub of

J

is not necessarily positive.

However,

forfixed a,ifb is sufficiently

large

and

(H4)

issatisfied, thenwecanprove Ub is positive. To make the proofmore transparent, we begin with the study oftwo simple cases.

LEMMA

5.2.

_Iff

_satisfies

thefollowing:

(H5)(i)

f(0)

_->_0;

(ii)

Thereexists

>

0 such that

f(a)

0 and

f(u)

>

0 in

(0, ), then,

wehave the following results:

(i)

Thereexists

b*

>

a such that

_for

anyb

>

b*, (2.1), (2.2)

has a positive solution

Ub that is alsoa localminimizer

_of

J(u)

over

H((a,

b)). Moreover,

(5.7)

0<Ub<

in(a,b),

and

(5.8)

bnF(a)<-J(Ub)<-

F(a)+C(bn-l+l)

for

somepositive constantCwhich is independent

_of

b.

(ii)

If

f(O)

> O,

then thereexists apositive solution

_of

(3.6),

(3.7)

and isstrictly

increasingin

(a, o)

and

(r)

as r-

.

Proof.

We

first modify the function outside

[0, ]

as C16mentand Sweers did in

[6].

Denote

0

u>=,

(5.9)

fl(u)

f(u)

u[0,

fi],

2f(0)-f(-u)

if u

<

0,

fo

f’

{

’2(r)

F,(u(r))}

dr.

FI(u)=

f(t)

dt and

Jl(U)--

rn-1 U

It

is easyto verify that

(5.10)

2f(o)lul

foru<0.

(12)

Hence,

forany u

H((a,

b)),

(5.11)

Since

f(u)

is

bounded,

the minimizer of

J1

(U)

over

H((a,

b))

is achieved, say

Ub, which is a solution of

(2.1),

(2.2).

By

(5.11),

Ub can be chosento be nonnegative. IfUb 0 in

(a,

b),

then by

Lemma 2.3,

Ub

>

0 in

(a, b).

If

f(0) >

0, then

Ub>O

in

(a,

b).

If

f(0)

0,we wantto provethat

Ub>O

in

(a, b)

if b is large enough. This can be done by choosing appropriate test functions

Ub*

H((a,

b))

asfollows"

I-a)fi

forr[a,a+l],

(5.12)

U*b(r)

for re

[a

+

1, b-

1],

[(b-r)a

forr[b-l,b].

Then

(5.13)

Jl(Ubg)

<--

_F(fi)+

_{C(b-I +}

₁₎

n

for some constant C which is independent of b.

Therefore,

ifb is large enough, then

Ub>O

in

(a,

b),

and by

Lemma 5.1(i),

Ub <fi in

(a,

b). By

(5.13),

(5.8)

follows. This proves

(i).

To

prove

(ii),

we first note that

f(0)>

0 implies there exists

_a.>0

such that

(0,

a,)

N and

(5.14)

lim

b(a)=

a

cO

(see

Lin

[18]).

On the other

hand,

by

Lemma 5.1(ii),

there exists

a*>

_a.

such that

U

re(a*, c).

Therefore,

for any

b>b*,

there exists

a(b) (0,

a*]

such that

u(., a(b))

isaminimizerof

Jl(u).

Therefore,

by

(5.14)

there exists

c

>

0 and asequence

bk

such that

a(bk)

ask.

By Lemma 2.4, tP

and

u(., c)

satisfies

(M).

By

Lemma

5.1(i)

and

(H5)(ii),

u(-, c)

is strictly increasing in

(a, ).

Thisproves

(ii).

Next,

we treatthe case

f(0)<

0.

LEMMA

5.3.

_Iff

satisfies

thefollowingconditions"

(US)’(i)

f(0)

_-<_0;

(ii)

There exist

>

u

>

0 such that

_f

u_)

_f

0 and

_f

u

<

0 in

(0, u)

and

f(u)

>

0 in

(u__,

(iii)

f(u)

du

>

O,

then there exists

b*>

a such that

_for

any b

> b*

there exists apositive solution Ub

of

(2.1),

(2.2)

with Ub

(%

),

where

io’

(5.15)

f(u)

du O.

Proof.

If

f(0)=

0, then the argumentsin

Lemma

5.2also work and give theresult as in

Lemma 5.2(i). Note

that

(H5)’(iii)

implies

Jl(Ub*)<0

in

(5.13)

when b is large

enough.

If

f(0)<

0, then

(5.10)

implies the extension

_fl

in

(5.9)

is no longer suitable to

the minimizationproblem.

Therefore,

we want to modify

f

in a differentwayand use

super- and subsolution methods to obtain solutions for

(2.1),

(2.2).

Since

f(0)<

0,we can extend

_f

into

(Uo,

0)

such that

(5.16)

f(uo)=O,f(u)<O

in

(Uo, 0)

(13)

and

(5.17)

’c’f(u)du

>

O.

uo

Let

v u Uo anddenote /31

___-

U0and v2 t Uo.

Let

g(v)

=f(u)

Then

g(0)

g(/31)

g(/32)

0,

g(v)<0

in(0,/31)

and

We

then extend g outside

[0,

v2]

by making

and

Denote

Then,

as in

(5.10),

in

[0, v2].

g(v)>0

g(v)=0

forv>v2

g(v)

-g(-v)

forv<0.

G( v)

g(

t)

dt.

G(]v I)=G(v)

for allv<O and

(5.17)

canbe rewritten as

(5.18)

Define

(5.19)

in

(/31,

G(v2)

g(t)

dt=

f(u)

du>O.

.(/3)--

rn-1

v’2(r)-

G(v(r))

dr n-1

(5.21)

v"(r)+

v’( r)

+

g(

v(

r)

=0 in

(a,

b),

(5.22)

v(a)=O=v(b)

have apositive solution /3b which is also a minimizer of

J

with

b

J(

/3b)

<=----

G(/32)

+

C( bn-l

+

1)

and

(5.23)

Uo,

v2).

for some positive constant Cwhich isindependentof b.

Therefore,

there exists

b*>

a

such that for any b

>

b*,

the equations

b"

(5.20)

J(v*)

<=---

G(v2)

+

C(b"-I

+

1)

n

in

H((a,

b)).

Let

_v*

be defined as

_u*

in

(5.12),

butreplace ff by v2. Then

(14)

RADIAL SOLUTIONS ON WIDE ANNULI 1513

Let

/b--U0

--

Vb; then

and

Aab+f(ab)=Avb+g(vb)=O

in

a<[x[<b,

ffb=Uo<0

on[x

_I=a

and

_Ix[=b.

Hence,

t

is a subsolution of

(2.1),

(2.2).

Since is a supersolution and fi>

,

by

monotone iteration scheme

(see,

e.g.,

[11],

[18]),

there is a positive solution

_u

of

(2.1),

(2.2)

and ub satisfies

ff

<

u

<

,

which also implies

Ilu ll

(, ).

Now,

we can prove thefollowing multiplicity result forgeneral case.

THEOREM 5.4.

Assume

condition

(H4)

is

_satisfied.

Then,

we have the following

results"

(i)

If

f(O)

>-

_O,

then thereexists

b* <

asuch that

_for

anyb

>

b*,

(2.1),

(2.2)

has at

least2m 1 positive solutions

1,

u.2,

t2,

,

.Urn,

,

with

k-

<

and

.u

<

in

(a,

b),

and

_.u

_{II, 7}

(, )

for

k 2,...,m.

(ii)

If

f(O)<

O,

then thereexists

b*>

asuch that

_for

any b

>

b*, (2.1), (2.2)

hasat

least m positive solutions

a

<...

<

a,

with

_{Ila ll}

_for

k-

,

m.

Proof.

We

shall prove the theorem byinduction on m.

For

m 1, the results were proved in

Lemmas

5.2 and 5.3 under the conditions

(H5)

and

(H5)’,

respectively. The argumentsused inthelast two lemmas are also valid

for general cases; thus the details are omitted.

Wefirst studythe case

f(0)=>

0.

For

j 2,.

.,

m, denote

If0(u)

foru

[0, aj],

f(u)=

foru

[aj, c),

(2f(O)-f(-u)

foru<0,

and

F(u)

f(t)

dt,

J(u)

rn-’

u’2(r)-F(u(r))

dr.

It

is clearthat

_f/

is an extension of

_f

and

(5.24)

if

_Ilull

<

.,

then

J+(u)= J(u)

for j=1,.-.,m-1.

Assume

m=j(>-l) is true. Then there exists a

_b>

a such that forany b

>

b,

J(u)

has a minimizer

_.b

which is apositive solution of

(2.1),

(2.2)

and satisfies

(5.25)

and

b

(5.26)

b"

_F

()

-<

_J

_(uj,)

_-<

_F

_(tTj)

₊

_C

_{(b"-’ +}

1)

for some positive constant

_C

thatis independent of b.

Let

_U+.b

be as in

(5.12)

with fi replaced by

_+1.

Then

(5.27)

Jj+l(/,/+l,b)

----

Fj+I(/j+I)

--

Cj+,(b"-’+

1)

n

(15)

for some positive constant Cj+ independent of b.

Therefore,

by

(5.24)--(5.27),

there exists

/7+1->

_b,

such that for any b

_>/7/1,

minimizer

tj+l.b

of

J/l

satisfies

Jj

+

tlj

+ b

< Jj

lj,

b

Hence,

tj+,,b

#

_t,b

and

IItT+,,ll

(j+,,/j+l)"

Let

Nj+

(o

tE

N:

_’j+l

<

Ilull

</j+l.

Then

.+

is an open set and nonempty according to the last paragraph.

Therefore,

by Lemma 5.1, there existtwo positive numbers

c+

>

a+l, such that

(a__+,

aj+)

N+I,

a+lZ

N+I

and

c+1

N+I.

By

Lemma

2.4,

a+

and

c+1

belongto P.

Hence,

lim

b(a)=o=

lim

b(a),

(_j+)+

(j+)-andthenboth

u(., aj+)

and

u(.,

c+)

satisfy

(M).

Therefore,

there exists

_b+

_>-

_g+l,

such that for any b

>

b7+1,

(2.1),

(2.2)

has at least two positive solutions having

maximumvaluein

(y+l,

a+l).

Since

_a+l

is a supersolution, there existsthemaximum

positive solution

tj+l

of

(2.1), (2.2)

havingmaximum value in

_(y+l,

_+1).

This proves

(i).

Condition

(ii)

canbeproved by usingthearguments usedin

(i)

and

Lemma

5.3;

thus details are omitted. I3

In

the proof oflast

theorem,

we obtain the following results for

(3.6),

(3.7).

COROLLARY 5.5.

Assume

condition

(H4)

is

_satisfied.

Then,

we have thefollowing

results"

(i)

If

f(O)

> O,

then thereexistatleast 2m 1 positive solutions

ll,

2,

12,

lm

of

(3.6),

(3.7)

and

a(r)

-

as r--> o

_for

j=1,’’., m.

(ii)

If

f(O)<=

O,

then thereexist atleast 2m- 2 positive solutionsu.2,

,"

",

,

of

(3.6),

(3.7)

and

6(r)-->

asr-->o

_for

j=2,..., m.

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