Nodal solutions of nonlinear elliptic equations in annular domains

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NODAL SOLUTIONS OF NONLINEAR ELLIPTIC EQUATIONS IN ANNULAR DOMAINS

SOON YEUN JANG1 AND DAE HYEON PAHK2

ABSTRACT. We investigate the existence of radial nodal solutions of the elliptic equation !::>.U+h(lxl)f(u) = 0, in annular domains.

It is proved that for each integer k 2: 1, there exist at least one radially symmetric solution which has exactly k nodes.

1. Introduction

In this paper we consider the existence of radial solutions, which have exactly k-nodes for any given integer k, of the equation

(1.1) 6.u+h(lxl)f(u) = 0

u=O

in 11(a, b), on a11,

where11(a, b) = {x E ~nIa< Ixl <b},n ~ 2 and the functions f and h satisfy :

(AO) (AI) (A2) (A3)

hE C¹((0,00)), h(r) >⁰ for r >⁰

f E C1(~), u· f(u) ~ 0 lim f(u) = 00

u-too U

lim f(u) = O.

u-tO U

Received December 19, 1997. Revised March 12, 1998.

1991 Mathematics Subject Classification: 35B05, 35J60.

Key words and phrases: radial solutions, nodes, shooting methods, annular domain.

1Supported by the '96 Post Doc Program of the Korean Research Foundation.

2Supported in part by BSRI-97-1421, KOSEF-95-070l-04-0l-3 and KOSEF- GARC-98.

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This model was studied by several authors (see e.g. [1}..v[8]). The existence of positive solutions of (1.1) with various nonlinearities was studied by Bandle, CofIman and Marcus [1], Coffman [2], Garazier [3], Lin [6], and Ni and Nussbaum [7]. Also the uniqueness of solutions of this model was studied by Kwong and Zhang[5] and Ni and Nussbaum

[7]. .

Our paper is motivated by the recent work of Lin [6] who proved the existence of positive solutions of(1.1) on an annulus. We are interested in the existence of solutions of (1.1) which change sign. We are going to show that, for each integer k > 0, there exists a radial solution of (1.1) which changes sign exactly k times. The method used here is the backward shooting method combined with the Sturm comparision theorem after a suitable change of variables.

Our main result isstated as follows :

THEOREM. For each positive integer k, (1.1) hasatleast one radial solution which has exactly k-nodes.

2. Proof ofTheorem

Since we are interested ina radial solution u = u(r), we shall write (1.1) in the form

(2.0) n-1

u"+--u'+h(r)f(u) = 0 in (a,b).

r

Thus, forn 2': 3, in terms of variables

s = {(n - 2)r(n-2)}^-1 and u(s) =u(r) (2.0) can be rewritten as

(2.1)

where

u"(s)+^p(s)f(u(s» =⁰ ⁱⁿ ^{(s*, s*),}

2n-2 k= n-2'

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As for n = 2, in terms of variables

s= "21 -loga+logr and u(s) = u(r), Eq. (2.0) can also be written as (2.1) with

and s* = "21-loga+10gb.

Using backward shooting method, we consider the family of solutions of the initial value problem

(2.2) u"(s)+p(s)f(u(s)) = 0, for s< s*, u(s*) = 0, u'(s*) = -I, where / >

°

is the shooting parameter.

Here s* >

°

will be kept fixed throughout the paper.

For every / > 0, the problem (2.2) has a unique solution u(·) ==

u(-, /) with the maximal domain of definition (sb), s*).

It iseasy to check that (2.2) is equivalent to the integral equation

S*

(2.3) u(s) = /(s* - s)

-1

(t - s)p(t)j(u(t))dt, for s<s*

and the solution u also satisfies

(2.4) u(s) = u(s)+u'(s)(s - s)+

is

(t - s)p(t)j(u(t))dt, for sand sE (sb), s*).

From (2.3), ifu is positive in some interval (a, s*) witha _~ 0, then

(2.5) u(s) ~ /(s* - s) in (a, s*).

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ITu have k zeros in (s(T) , s*), we set

SI(,) = inf{SI Iu(s,,) > 0 in (SI, s*)}, s2(T) = inf{s2 < SI Iu(s,,) <0 in (S2,SI)},

By standard results in ordinary differential equations, the function (s,,) ^{- 7}u(s,,) and (s,,) ^{- 7} u'(s,,) are continuously differentiable in the set

((s,,) I,>0 and sE (s(T), s*)}.

SinceU'(Si(T),,) ⁼¹⁼O,i = 1,2,···k, by the implicit function theorem, the sets

11== {, >0 Isl(T) >O},

12== {,> 0 Is2(T) > O},

1k

={,

^> ⁰^I^sk(T) ^> ^O}

are open and Si(·) E C¹(li ), ⁱ ⁼ ^1,2,···^k. ^Clearly ¹^j ^C 1i, Sj(T) <

si(T), forbE 1j ,j >i.

Consider the sets

Jl = {,> 0 Iu'(r,,) = 0 for some r E (O,s*), andu(s,,) > 0 in (r,s*)}, J2

={,

^>⁰ Îû'(r,,) ⁼ ^{0 for some}^r Ê ^(O,sl(T)),

and u(s,,) < 0 in (r,sl(T))},

Jk == {, > 0 Iu'(r,,) = 0 for somer E (O,Sk-l(T)),

and (-l)k u (s,,) >0 in (r, Sk-l(T))}.

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If,E Ji andU'(T,,) = 0, then

(2.6) f(U(T,,)) f= O.

For, if f(U(T,,)) =0, then the initial value problem vl/(s) +p(s)f(v(s)) = 0 in (T, Si-1(,)),

V(T) =U(T,,), V'(T) =0

has a solutionv(s) = constant=u(r,1') for anys E (0,s*). Therefore, the uniqueness of initial value problem ofO. D. E. implies u(s,,) =

u(r,1') for anysE (0,s*), a contradiction. By (Al) and (2.6), if, E Ji andU'(Ti,') = 0 for someTi E (0,si-1b)), then for i odd

u'(s,,) < 0 for s E (Ti,si-1b)), u'(s,,) > 0 for s E (S(-Y),Ti) and if i is even

U'(s,,) > 0 for sE (Ti,Si-1(-Y)), u'(s,,) < 0 for sE (S(-Y),Ti),

where s(-y) = Si(-Y) if, E Ii, s(-y) ⁼ ⁰ ^if^l' rt. h Therefore, we shall denote this uniqueTi byTi (,)which is also the maximum (or minimum) point ofUhf) in (S(-y),Si-1(-Y)). It can be verified that Ji are open sets andTi(-) E CO(Ji).

LEMMA 2.1. Assume that the conditions (AO), (Al) and (A3) are satisfied. Suppose that for somei E N,

lim ri(-Y) = s*,

-y-+=

{

(X), ifi isodd, lim U(Ti(-Y),1') = if' .

-y-+= -00, Z 18even

and

lim Sib) = s*.

-y-+=

Then (2.7)

{

(X), ifi is odd, lim u' (Si (-y) ,1')= 'f' .

-y-+= -00, 1 Z IS even.

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Proof. Ifnot, there exist a constant M >0 and a sequence'Yk ---7 00

such that (-I)i+l u'(Si("(k), 'Yk) ~ M. By (2.6)

0< (-I)i+ 1u(Tib'k)) = (-I)i+l{u(Sib'k))+U'(Sib'k))(Ti('Yk) - sib'k))

I

^{Ti ('Yk)}

+ (t - Ti("(k))p(t)j(u(t))dt}

Si ('1'10)

= (-I)i+1 u'(Sib'k))(Ti("(k) - Si("(k))

l

^{Ti ('Yk)}

- (_1)i+¹ (Ti("(k) - t)p(t)j(u(t))dt

sibk)

~ M(Ti("(k) - Si("(k)),

since (_1)i+¹f:''c~:i(Tib'k) - t)p(t)j(u(t))dt :;:::O. By the assumption, (Tib'k) - Si("(k)) ---70 ask ---7 00, which is impossible. D

LEMMA 2.2. Assume that the conditions (AO), (AI) and (A2) are satisfied. Then Ti("() ,i

=

^{1,··· and} Si("() ,i

=

1,··· are well defined when'Y issufIiciently large and

(2.8) (2.9)

(2.10)

lim Si("() =s* i =1,· .. ,

'Y-too

lim Tib') =s* i =1,· .. ,

'Y-too

{

00, ifi is odd, lim U(Ti("(),'Y) = if ..

'Y-too -00, ZIS even.

Proof. By induction, _~-tooTl ("() = ^S*,~-too u(Tl ('Y), 'Y) = ⁰⁰

and ~-tooSIb) = s*, by lemma 2.1 and lemma 2.2 of [6].

Assume that

lim Ti("() = s*,

'Y-too

lim Si("() = s*,

'Y-too

{

00, ifi is odd, lim U(Ti("(),'Y) = if ..

'Y-too -00, Z 18even .

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We must show that (2.11)

(2.12)

(i) (ii)

lim Ti+lb) =^s*,

'"(--+00

lim si+lb) = s*,

'"(--+00

(2.13) (iii) _{ ^-00, if i is odd ,

lim U(Ti+lbky) = if ..

'"(--+00 00, Z 18even .

(2.14)

(i) Ifnot, there exist TO E (0,s*) and a sequence'Yk - 00 such that Ti+l bk) ~TO for all k.

Lets=

TotS

^{* •} Then, by induction hypothesis, for every sufficiently large k, Si('Yk) >TO and

(-l)iu(s,'Yk) > 0

(-l)i u'(s,'Yk) ~ 0 for S^E (TO,S).

We now claim that

(2.15) limsup(-l)iu(s,'Yk) = 00.

k--+oo

Suppose that this is not the case. Then there exists a constant M > 0 such that (-l)iu(s,'Yk) ~ M for all k. Now, by (2.4) and lemma 2.1,

M _~ (-l)iu(s,'Yk)

= (_l)i{u(sibk), 'Yk)+u'(sibk), 'Yk)(S - Si('Yk))

+

1

⁸ (t - s)p(t)j(u(t))dt}

Si('"(k)

= (-l)i-lU'(Sibk), 'Yk)(Sibk) - s)

l

^{Sibk )}

- (_l)i 8 (t - s)p(t)j(u(t))dt (2.16)

~ ^(-l)i-l^U'(Sibk),'Yk) (S*; s) _ C for some constant C.

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By Lemma 2.1, the right side of (2.16) goes to ^00. Itis a contradiction. Thus (2.15) holds. By choosing a subsequence of'Yk if necessary, we may assume,

(2.17)

By(AO) ,thereisa subinterval(0:,(3) C (ro,s)such thatpes) ~ Po >

o

ⁱⁿ ^(0:,(3). ^Set

Then

Mk ~inf{!(u) Ilul_u ~ ^(-l)i^u^(s,'Yk)}.

By (2.17) and (A2),

(2.18) lim Mk = 00.

k-too

By (2.2),Uk

=

^{u(·, 'Yk)} ^satisfies

u%(s) +

p(s)f~:(~~))

^Uk(S) ⁼ ⁰ ⁱⁿ ^(0:,(3),

where

(2.19) ( )!(U(S,'Yk)) > M . ( (3)

p s ( ) _ PO k ill 0:, . U S,'Yk

Now, letVk be a solution of

V"+^POMkV⁼ 0 in (0:,(3).

By (2.18) , Vk has at least two zeros in (0:,(3) when k is sufficiently large. By (2.19) and Sturm Comparison Theorem, Uk has at least one zero in (0:,(3). Because of (2.14), this is impossible. Hence (i) holds.

Now, let's show (ii). Ifit is false, there exist a pointSo E (0,s*) and a sequence'Yk such that'Yk ----? 00 with Si+! ('Yk) :::; So for allk, and

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(2.20)

(-l)iu(s,'Yk) > 0 and (-l)iu'(s,'Yk) _~ 0 in (So,Ti+IC'Yk)).

Set s =

sots

^*. By (i), we may assumes < Ti+IC'Yk) for sufficiently largek. We claim that

(2.21) limsup(-l)iu(s,'Yk) < 00.

k-too

Otherwise, there exists a subsequence'Yk -+00 such that

Similar to (i), by Sturm Comparison Theorem, Uk == U(·,'Yk) has zeros in (s, Ti+IC'Yk)) when k is sufficiently large, which is impossible by (2.20). Hence (2.21) holds. By (2.4)

(-l)iu(s,'Yk)

= (_l)i{U(SiC'Yk), 'Yk)+U'(SiC'Yk),'Yk)(S - SiC'Yk))

+

l

^s (t - s)p(t)f(u(t))dt}

sib'k)

= (-l)iu'(si('Yk),'Yk)(s - SiC'Yk)) fSi<'rk)

- (_l)i

ls

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

;:::: (-l)i+I U'(SiC'Yk), 'Yk)(SiC'Yk) - s) - C.

By Lemma 2.1. and (2.21), it is impossible. Hence (ll) is true.

(iii) Suppose it does not hold. Then there exist a constant M > 0 and a sequence'Yk -+ 00such that

(2.22) Set

F(u) =

l

^u ^f(s)ds

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and define

Since

V(s) ==V(s, 1') =2I [u'(s)]2 +p(s)F(u(s)).

(2.23)

V'(s) = u'(s)u"(s)+p(s)f(u(s))u'(s)+p'(s)F(u(s))

= p'(s)F(u(s)),

Therefore, we have

2I [U'(SiC'Yk))]2 +p(SiC'Yk))F(u(SiC'Yk))

= ~[u'(^Ti+l^('Yk))]2 +p(Ti+lC'Yk))F(U(THIC'Yk))

1

^{8i (-rk)}

+ p'(t)F(u(t))dt,

'TH1 (-rk)

~[u' (SiC'Yk))]2 = p(Ti+lC'Yk) )F(u(THlC'Yk))

l

^si^(-rk)

+ p'(t)F(u(t))dt.

'Ti+1 (-rk)

(2.22) implies that the right side of (2.23) is bounded. Bylemma 2.1.,

it is impossible. Therefore (iii) holds. 0

LEMMA 2.3. Assume that the conditions (AO), (AI) and (A3) are satisfied. Then wehave

(2.24)

(2.25)

(i) if (O,'YI) C J1 ^for ^some 1'1 >0, then limTIC'Y) = 0.

"Y--+O

(ii) if (0,1'1) C It forsome 1'1 > 0, then limSI (1') = o.

"Y--+O

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Proof See [6]. 0

LEMMA2.4. Assume that the conditions(AO), (AI), (A2)and(A3) aresatisned. Then we have"

('Yil,'Yi2) ofIi , i = 1,2,·,· . (2.26)

(2.27)

(i) for anyconnected component lim Sih') = 0 'Y-+'fti

(ii) for any connected component ('Yib'Yi2) ofJi ,

lim Ti(,) = 0 i = 1,2,··· . 'Y-+'fti

Proof (i) Let('Yil''Yi2) be a connected component ofIi (i⁼ 1, 2, ... ).

Either'Yil =0or'Yil > O. If'Yil = 0,by lemma 2.3. (ii), lim'Y-+'ftiSI ('Y)

= O. SinceIi chandsIh') >Sih')for eachi = 2,3,···, (2.26)follows from (2.25).

If'Yil > 0 and suppose that this is not the case, there exist a point Si > 0 and a sequence'Yik ^--+'Yil withSi(,ik) ^--+ Si, i = 1,2,· ... Then

Le. 'Yil El, which is a contradiction to the fact that ('Yil,'Yi2) is a connected component ofI_{i .}

(ii)Next, let ('Yil,'Yi2)be a connected component ofJi, (i ⁼ 1,2,··· ).

If 'Yil = 0, the result follows from lemma 2.3. If 'Yil > 0 and lim'Y-+'ftiTi('Y) ⁼¹⁼ 0, then there exist aTi > 0 and a sequence'Yik ^--+ 'Yil withTih'ik) --+ Ti as k --+ 00. Since

U'(Ti,'Yil) = lim U'(Tih'ik)"ik) =0

k-+oo

Le. 'Yil EIi , which is a contradiction. o

Now, we prove the main Theorem.

THOREM 2.5. Assume that the conditions (AO), (AI), (A2) and (A3) are satisned. Then for each integer kEN, (1.1) has at least one nodal solution which has exactly k nodes for all a, b such that 0< a< b< ^00.

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Proof. Fix kEN. By lemma 2.2, h+l =1= 4> and there exists '7~ 0 such that ('7,00) C 1k+l. lemma 2.4 implies

Hence for any s* < s*, there exist 7* > '7 such that sk+lb*) = s*.

Sincesib) > sk+lb) for 0 ::;i <k,there exists a nodal solution which have nodes

D

References

[1] C. Bandle, C. V. Coffman, and M. Marcus, Nonlinear elliptic problems in annular domains, J. Differential Equations 69 (1987),332-345.

[2] C.V. Coffman, A nonlinear boundary value problem with many positive solu- tions, J. Differential Equations 54 (1984),429-437.

[3] X. Garazier, Existence of positive radial solutions for semilinear elliptic equa- tions in the annulus, J. Differential Equations 70 (1987),429-437.

[4] R. Kajikiya, Existence and asymptotic behavior of nodal solutions for semi- linear elliptic equations, J. Differential Equations 106 (1993), 238-256.

[5] M.K.Kwong andL.Zhang, Uniqueness of the positive solution of .6.u+ f(u) =

oin an annulus, Differential Integral Equations 4 (1991), 583-596.

[6] S. S. Lin, On the existence of positive radial solutions fornonlinear elliptic equations in annular domain, J. Differential Equations 81 (1989), 221-233.

[7] W.-M. Ni andR. D. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of .6.u+f(r, u) = 0, Comm. Pure Appl. Math. 38 (1985), 69-108.

[8] W.-M. Ni, Uniqueness of solutions of nonlinear Dirichlet problems, J. Differ- ential Equations 50 (1983), 289-304.

Department of Mathematics Yonsei University

Seoul 120-749, Korea