Multiplicity results of ordered positive solutions for semilinear elliptic problems on $R^n$

MULTIPLICITY RESULTS OF ORDERED POSITIVE SOLUTIONS FOR SEMILINEAR

ELLIPTIC PROBLEMS ON Rn

BONGSOO Kot AND YONG-HOON LEEt

ABSTRACT. We prove the existence of2N -1 distinct ordered pos- itive solutions of a class of semilinear elliptic DiricWet boundary value problems onRn when the forcing term hasN distinct posi- tive stable zeros and the coefficient function decaying to the zero at infinity.

1. Introduction

Inthis paper, we are concerned with the existence of2N -1 distinct ordered positive solutions of the following semilinear elliptic boundary value problem;

(P.x) _{

D..u+Ag(lxl)f(u) =

°

lim u(x) = 0, Ixl-too

where n ~ 3. In what follows, we assume f E C(I,R), 1= (O,b] eR is a bounded closed interval, and9 E C(R+, R+), g(r) >

°

r ~ 0, denoting R+ = (0,00). Furthermore, we assume the following conditions;

Received September 21, 1998.

1991 Mathematics Subject Classification: 35J25, 34B15.

Key words and phrases: existence, multiplicity, positive solution, semilinear el- liptic problem, singular boundary value problem, upper and lower solution method, three solution theorem, Schauder fixed point.

tThis work was supported in part by Cheju National University, Research Fund, 1997.

*

(fo) There exists a positive constant M such that feu) - f(v) ~ -M(u - v), ifU,v E I, u ~ v.

(h) There exist exactlyN positive numbers 0<al < a2 < ... < aN such that f(ai) = 0, for alli = 1,··· ,N.

(h) [0,aN] C I and there exists a positive constant Kt such that

!'(ai) _~-Kf, for alli =1,··· ,N.

(fs) Isai ^{f( u)du >0, for all S E [0,ai)} and for alli =1, ... ,N.

(g) 9 E C1(R+) satisfies It^{rg(r)dr < 00.}

If the domain of the equation in problem (P>.) is bounded and open with smooth boundary, then there have been several studies ([2],[3],[4],[5],[7],[8]) which prove generally that there exists Ao >

°

such that (P>.) has at least 2N - 1distinct ordered positive solutions for allA> Ao •

In those studies, the boundedness of the domain is crucial. Ifthe domain is unbounded, the situation is quite different because the com- pactness of operator or functional induced from the problem is not guaranteed. Thus it is interesting to study the existence of multiple positive solutions when the problem is defined onRn, and as far as the authors know, related studies have not been made yet.

To obtain our desired result, we split Rn into a ball B(O,l) and its compliment n = Rn \ B(O, 1) and solve problem (P>.) defined on B(O,l) and nrespectively.

As mentioned above, we get the existence of N positive ordered solutions for (P>.) defined onB(O,1) ifAis sufficiently large.

To obtain the existence ofN positive solutions for the problem de- fined on the exterior domain n, we transform (1).) into the singular boundary value problem (S>.): More precisely, the problem on 0, we concern, is as follows;

(1).)

{

L}.u+Ag(lxl)f(u) =0 u(x) = aj if Ixl⁼ 1,

lim u(x)= 0.

Ixl-+oo

inn,

By transformationsr = Ixl andt =r^{2- n,} (1).) can be transformed into

{

u"+Aq(t)f(u) =0, 0 <t <1, (S>.)

u(O) =^{0, u(l)} =^aj,

where qis given by

1 ^{2(1-n) 1}

q(t) = t ^n-2 g(t^{2 - n ).}

(n - 2)2

It is known that a positive solution of (S>.) corresponds to a positive radial solution of (1).).

In the problem (S>.), the coefficient function q is of two types de- pending on the asymptotic growth condition on 9 as follows; if9 in (P>.) satisfies limr-toor 2(n-l)g(r) < ^00, then q can be extended con- tinuously on [0,1], so problem (S>.) is regular. On the other hand, if 9 satisfies limr-toor 2(n-l)g(r) = 00, then q E C(O,l] and singular at t = 0, see [9] for further detail.

For problem (S>.),we show the existence of positive solutions for the singular case. We know the result for the regular case is followed by that of the singular case with less conditions.

Once positive solutions of two problems are provided, we glue them together to get N-pairs of lower and upper solutions of(P>.), and then we use Three Solution Theorem ([1],[8]) and an argument of some con- vergence method ([10]) for proving our multiplicity.

In Section 2, we give a fundamental existence theorem of positive solutions on the method of G-upper and G-Iower solutions which might have singular points in the interior of the interval [0,1]. In Section 3, we study the multiplicity of positive solutions for problem (P>.) in the process as the previous paragraphs.

2. Existence of solutions for singular boundary value prob- lems

In this section, we prove a fundamental existence theorem in terms of general upper and lower solutions for singular boundary value problems of the form (SBVP);

{

u"+f(t, u) = 0, 0<t <^1,

(SBVP) u(O) = A, u(l) = B,

wheref :D c (0,1) x R ^--tR is continuous. A solutionu(·) means a function u E C[O, 1]nC2(0, 1) such that (t, u(t)) ED, for allt E (0,1) and u satisfies the ordinary differential equation pointwise on (0,1) with u(O) =^{A andu(l)} =^B.

DEFINITION 1. a E C[O, 1] nC2(0, 1) is called a lower solution of (SBVP) if(t, aCt)) E D for allt E (0,1) and

a"(t) +f(t,a(t)) 2: 0, a(O) :s;A, a(l) :s; B.

Similarly,(3E C[O, 1]n^{C2(0, 1) is called an}upper solution of(SBVP) ifthe above inequalities are reversed.

The fundamental theorem on upper and lower solution method for the problem (SBVP) is studied by Habets and Zanolin [6]. We give definitions of somewhat general type of upper and lower solutions and prove a fundamental theorem for this type.

DEFINITION 2. We say that a continuous function a(·) :[0,1] ^--tR is a G-lower solution of(SBVP) ifa E C²(0,1) except at finite points

Tb . .. ,Tn with

°

(L1) at each Ti' there exist a'(Ti-), a'(Ti+) such that a'(Ti-) <

a'(Ti+) and

{

a"(t)+f(t,a(t)) 2:0, for alltE(O,l)\{Tb···,Tn },

a(O) :s; A, a(l) :s;B.

We also say that a continuous function (3(.) : [0, 1] ^--t R is a G-upper solution of(BBVP) if(3 EC²(0,1) except at finite pointsUb··· ,Um

with

°

{

(3"(t) +f(t,(3(t)) :s; 0, for all t ^E (0,1) \ {U1,··· ,um}, (3(0) 2:A, (3(1) 2:B.

REMARK 1. We note that lower and upper solutions of (SBVP) imply G-Iower and G-upper solutions, respectively. But not vice versa.

Hence, Theorem 1 below is a generalization of Theorem 1 in [6].

DEFINITION 3. Ifa, f3 E C[O,1] are such that a(t) ~ f3(t), for all t E [0, 1],we define the set

D~= {(t, u) E (0,1) x R: a(t) _~ u _~f3(t)}.

THEOREM 1. Let a andf3 be, respectively, a G-lower solution and a G-upper solution of(SBVP) such that

(a1) a(t) _~ f3(t), for allt E [0,1].

(a2) _D~ cD.

Assume also that thereisa function h E C ((0,1),R+) such that (a3) If(t, u)/ _~h(t), for all(t, u) E D~ and

(a4) 10¹s(1 - s)h(s)ds< ^00.

Then (SBVP) has at least one solution u such that

a(t) ~u(t) ~f3(t), for allt E (0,1).

Proof. Define a modified function off asfollows;

f(t,f3(t» - u

1-_+uf3~)^{, ifu >f3(t),}

F(t, u) = f(t, u), ifa(t) _~u _~ f3(t), u - a(t)

f(t, a(t» - 1+u_{2 '} ifu< a(t).

ThenF :(0,1) x R ^--tR is continuous and

(1) IF(t,u)1 _~ m(a,f3)+h(t),

for all (t,u) E (0,1) x R, where m(a,f3)=^{11a 1100}+1If31100+1.

Consider the problem

(2) {

u"+F(t,u) = 0, 0< t < 1, u(O) =A, u(1) =B.

We claim that any solution uof (2) satisfies o:(t) ~u(t) ~(3(t), for all t E [0,1]. Without loss of generality, we only prove thato:(t) ~ u(t)for allt E [0,1]. We can prove for the case thatu(t) ~(3(t) for allt E [0,1]

by a similar fashion.

Suppose that 0: t=. u. So let (0: - u)(to)= maxtE[O,lj(O: - u)(t) > O.

Ifto E (0,1) \ {Tb'" ,Tn},^then (0: - u)"(to) ~O. Sinceu(to)< _o:(to),

o~ (0: - u)"(to)=o:"(to)+F(to,u(to))

"( ) f( ()) u(to) - o:(to)

= 0: to + ^{to, 0: to - 1}+u to2()

> o:(to) - u(to) > 0 - 1+u²(t_{o )} ^, a contradiction.

Let to = ^{Ti, for some i = 1,'" ,n. Since} 0:-u attains its positive maximum at^Ti,

Thus

This leads to a contradiction to the definition of G-lower solution.

Let to = 0 or 1

0< (0: - u)(O) = 0:(0) - A ~ 0, 0< (0: - u)(1) =0:(0) - B _~ 0, a contradiction.

Therefore, o:(t) ~u(t) ~(3(t),and so we can concludeuisa solution of(SBVP).

We claim that (2) has at least one solution. It is well-known that problem(2) is equivalently written as

u=Tu onX =C[O,1},

where

Tu(t) = A+(B - A)t+

1

{

s(l - t), O::S; s::s; t, G(t s)=

, t(l-s), t::S;s::S;l.

By (1) and (a4), T : X - X is well-defined, continuous andTX is bounded. IfT is a compact operator, then the proof of the existence of a solution is done by Schauder fixed point theorem.

To showT compact, making use of Arzela-Ascoli Theorem, it suffices to show that T X is equicontinuous.

Let t E (0,1). Then by (1) we get 'dtTu(t)1d

::s; IB - AI+

1

1

::s; IB - AI+ m(~,^{(3)(t2+(1-t)2) +}

1

1

~ ^{IB - AI}+mea,^{(3)(t2+(1 - t)2) +}'"Y(t).

2

If'"Y E L¹(0, 1), then the proof follows from that

[1 1'"Y(s)lds::S; lim (1 - t)

r

10 ^t-.+1- 10 ^{t-.+O+ t}

+

21

41

D

0<t < 1,

REMARK 2. It is not hard to see that if we replace (a4) by the condition

I:

EXAMPLE. As an application of Theorem 1, we solve the following singular boundary value problem:

{

u"+AU (U -

:t)

u(O) =^0, u(l) =^O.

We can easily check that problem (3)..) satisfies (a3) and (a4) in Theo- rem 1. It seems to be quite difficult to find a pair of lower and upper solution for (3)..). Because, the functionu(t) = it intersects the func- tion u(t) = 1in the open interval (0,1). However, by Theorem 1,it is enough to construct a pair of G-Iower solution and G-upper solution to get a positive solutionu(t;A) of (3)..). The following theoremis useful to construct a pair of G-Iower solution and G-upper solution.

THEOREM. (Theorem 4 in [8]) Let J ^E G¹([a,b] x I,R). Suppose that there isa positive real numberU1 satisfying the following condi- tions:

(i) J(t,UI) = 0 for allt E [a,b],

(ii) Ju(t,U1) S _k² < 0 for allt E [a,b], and

(ill) Is^{u1J(t,u)du > 0 fort} =^{a andt} =^{b ands E [0,}UI).

Then thereisapositive numberA_osuch that for allA> Ao ' the problem

{

u"+^AJ(t,u) =0, u(a) =0=u(b)

a < t < b,

hasa positive lower solution17satisfyingpet) s^{U1 for allt E [a,bJ.}

Let J(t,u) =u (u - it) ^{(1-u). At t = 1,}we note that

1

for all s E [0,1). Hence, there exists a positive number 8 so that

1

:t)

uniformly on t E [1 - 0,1] and for all s E [0,1), and we can apply the above Theorem with U1 = 1 on the interval [1 - 0,1], and obtain a positive A_o so that for all A> A_{o '} the following problem:

{

u"+AU (U -

:t)

u(1- 0) =0, u(1)= 0.

1-0< t <^1,

has a positive lower solution vet;A) satisfying vet;A) ::; 1 for all t E

[1- 0,1]. ObviouslyU1 =1is an upper solution of(3i).

We define a G-Iower solution aCt;A) by

{

0, 0::; ^{t ::;}1 - ^0, a t·A -

(, ) - vet;A), 1 - <5 ::;t ::;^1.

Thena::; (3=1 are a G-Iower solution and a G-upper solution of (3_A), respectively. Consequently, by Theorem 1, there is a positive solution u(t;A) of (3A) which lies between aCt;A) and 1.

3. Ordered positive solutions

We prove the existence of2N - 1distinct ordered positive solutions of the following problem;

{

6 u+Ag(lxl)j(u) =

°

lim u(x) = 0.

Ixl-+oo

H the domain n of the equation in (P_{A )} is contained in the large ball B(O,r) with r > ro and has the smooth boundary, if the function 9 E G1(n) satisfies g(ro ) > 0, and if we also assume the conditions (fo) ^rv (h) for j, then the multiplicity result of positive solutions for large enough A can be obtained ([4],[8]).

REMARK 3. Necessary and sufficient condition that 9 satisfies the integral condition in (g) is that qfulfills the following condition;

(G)

1

which corresponds condition (a4) in Theorem 1. We also note that 9 satisfies condition in (g) if JRn\B(0,r)p(lxl)lxI2-ndx < 00 for some r > O. Furthermore, if we let hex) = ^g(lxl), we get the following statement: iffor somep with1<^p< ~, hEV(Rⁿ \ B(O,r»,then q satisfies the integral condition(C). But the converse may not be true.

This also implies that the nonlinear operator induced from(P_{A )} might not be compact.

The following is the main result in this paper.

THEOREM 2. Let n ~ 3 and assume (io) '" (13). If 9 satisfies con- dition (g), then thete exists Ao> 0 such that (P>..) has 2N -1 distinct ordered positive solutions forallA> Ao •

Proof Without loss of generality, we assume that there exists a point Xo such that Ixol < 1 andg(lxol) > O. Let

n={x E Rn: 1< Ixl< oo}.

Then the problem

{

6 u+Ag(lxl)f(u) = 0 u(x) = ^aj if Ixl =1,

lim u(x) = 0, Ixl-+oo

can be transformed into

inn,

(S>..)

{

u"+^{Aq(t)f(u) =} 0, u(O) = 0, u(l) =aj'

0<t <1,

We notice by condition (g) that Theorem 1 is valid for (S>..). Itis not hard to check that 0 andaj are a lower solution and an upper solution of (I>..), respectively, forallA, thus (S>..) has solutions Yj = Yj(tj A), j =

1,'" ,N with 0 :S Yj(t; A) :s aj, for all t E [0, I}, and thus (I>..) has positive radial solutionsilJ (x) = Yj(lxI2-njA), j = 1,'" ,N,for all A such that

o:s^ilJ(x) :Saj for all x E n ^{and lim} il~(x)= O.

Ixl-+oo J

On the other hand, since g(lxol) > 0, from Theorem 5 in [8], there existsA_o >0such that forall A>Ao ' the following problem

{

I:::.U+Ag(lxl)f(u) = 0 in 0::; Ixl <1, u(x) =0 if Ixl =1

has N distinct positive ordered solutions

v;

v;

for all 0 ::; Ixl ::; 1 and

v;

w;

{

a. if 0::; Ixl ::;1, w](x) = u~(x) ^{if Ixl} ~ ¹

and the function w] defined by

_>..( ) _ {V](X) ^if 0::; Ixl ::;1,

w· x -

J 0 if Ixl ~ 1

are a quasi-supersolution and a quasi-subsolution ([8]) respectively of the following problem;

{

l:::.u+Ag(I~l)f(u)=O u(x) =0 if Ixl= R.

in Ixl <R,

S·mcew^->..j -< w^->..j ' w^->..j + 1 -<Wj+l, Wj^{->.. ->..} -< w^->..j +l,andWj+l->.. <l=FWj'->.. Theorem3 in [9](or Three Solution Theorem in [1]) implies the existence of three distinct solutionsuf -<u~~ -< uf+l of(4R) such that

->.. < ^{R ->.. ->.. R ->..}

wj _ uj -<Wj' Wj+l -< uj + 1 -<Wj+l'

W](XR) < _uf+~(XR) < wJ-H(XR),

for some XR with IXRI < 1. Fix j and A, letting R - 00, Theorem 2.10 in [10] implies that, without loss of generality (otherwise using

diagonal process),

are solutions of problem (P>.) , and those are distinct since aj, tit+l

are independent of R and liminfR-+oo IXRI ::; 1. Trivially, they are positive. Since limlxl-+oouJ(x) =0, we get Uj(x) ---70 as Ixl ---700 and

this completes the proof. 0

From the simple scaling technique, we have the following result.

COROLLARY. With assumptions in Theorem 2, there are 2N - 1 distinct ordered positive solutions of the following problem for allsuf- ficiently largeoX:

{

lim u(x) =0, Ixl-+oo

REMARK 4. Let g(O) = 1. With the above corollary and some convergence method, we hope to prove the same multiplicity of positive ordered solutions for the problem ~u+f(u) = 0 inRn with the same limiting behavior at the infinity. But the problem is still open.

ACKNOWLEDGEMENT. The authors return thanks to the referee for several comments on this paper.

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Bongsoo Ko

Department of Mathematics Education Cheju National University

Cheju 690-756, Korea

E-mail: [email protected] Yong-Hoon Lee

Department of Mathematics Pusan National University Pusan 609-735, Korea

E-mail: [email protected]