Uniqueness results for a class of positone problems

Nonlinear Analysis, Theory. Printed in Great Britain.

UNIQUENESS

Methodr

C? Appliconons,

Vol. 7, No. 2. pp. 223-230.

RESULTS

FOR A CLASS OF POSITONE RATNASINGHAM

Department

0362-546X/83/020223SOS $03.00/O 0 1983 Pergamon Press Ltd

1983.

PROBLEMS

SHIVAJI*

of Mathematics, Heriot-Watt University, Edinburgh, U.K (Received in revised form 16 July 1982)

Key words and phrases: Nonlinear boundary value problems,

nonnegative solutions

1. INTRODUCTION CONSIDER the nonnegative

solutions -Au(x)

of the boundary = nf(u(x))

value problem for x E D, for x E dD,

u(x) = 0

where ?, 2 0, D is a bounded region in iw” with smooth is defined on [0, r) for some r, 0 < r G ~0 and satisfies:

boundary

(1.1)). dD and the nonlinearity

f

(fl) f E C2([0, r)) and f(u) > 0 for 0 4 u < r; (f2) there exists w 3 0 such that f(u) - f(u) 2 -w(u - U) for all u, u E [0, r) with u > v; (f3a) r = l tooand there exists Co > 0 such that f(u) 6 CO for all u 2 0 OR (f3b) r < + CQand fE C2([0, r]) with f(r) = 0. The above hypotheses ensure that (1.1)~ has a nonnegative solution for all L 3 0 (see [l]). In particular we proved in [l] that if f is sufficiently convex for small values of u then there exist values of A for which there exist at least three distinct nonnegative solutions (see also [2] for another type of multiplicity result for (1.1)~). H ere we shall discuss the uniqueness of nonnegative solutions for small and large values of A under some additional conditions. We provide further evidence that the bifurcation curves for some of the problems discussed in [l] are S-shaped. It was shown in [l] that hypotheses (fl), (f2) and (f3a) or (f3b) guarantee the existence of maximal and minimal nonnegative solutions denoted by fin and UI respectively of (1. l)*. We shall prove uniqueness by assuming that tin and UI are unequal and by obtaining a contradiction for small and large values of A. In Section 2 we obtain uniqueness results for small A. In Section 3 we obtain uniqueness results for large A when D = (0,l) C R by analysing the function H(U) =Ji f(t) dt - (u/2) f(u). In Section 4 we obtain uniqueness results for large il when, D = D, the unit open ball in LQ”;n 2 2 and hypotheses (f3a) is satisfied. We shall also discuss, where appropriate, how our results apply to the problem -Au(x)

= A exp(au/(cu+

u(x) = 0

* Present Address:

Department

u))

forx ED, for x E JD,

of Mathematics, Yarmouk University, Irbid, Jordan. 223

(1.2)A

224

R. SHIVAJI

where (Y> 0 which arises in the theory of combustion (see [ 11) and to the problem -Au(x)

= ~(cJ- u) exp(-k/(1

+ u))

forx ED,

U(X) = 0

for x E aD,

(1.3)A

where u and k are positive constants which arises in chemical reactor theory (see [l] and [3]). 2. CASE

WHEN

1 IS SMALL

In this section we obtain a uniqueness result for small values of A. Recall tik and UAand define WA(X):= ti&) - U*(X)in d. Assume WA(X)f 0. Then there exists a XLE D such that w&~) 2 We in D. It follows easily that WA(X)satisfies

I

wA(x) = A D G(x,

y>f(W)) - fht.~))

where G(x, y) is the Green’s function of the problem. z&(y)) such that

dY

inD,

Then there exists a 8*(y) E (u*(y),

I

WA(X)= A D G(x, r>f'(W)b4~>

dy

inD,

(2.1)

and so G(~A,y)f’(6(y))w&)

dY.

(2.2)

We will now show that (2.2) is violated for small A. We shall make use of the function z&(x) : =

ID

in D

G(x> Y) dy

(2.3)

where G(x, y) is the Green’s function of the problem (1.1)~ and the constant M : = s,ug uo(x). THEOREM

2.1. Let fsatisfy

(2.4)

(fl), (f2) and (f3a) or (f3b). Suppose there exists h > 0 such that

f’(u) s h

forOsu
(25)

Then (l.l)~, has a unique nonnegative solution for 0 < A < (hM)-‘. Proof. Since f’(u) G h for 0 < u < r, from (2.2) we have We

s Ah s

ID

G(~A, y)wiCy) dy s Ahwdx$

ID

G(xJ.,Y) dy = Ahw&duo(xd

hhwA(x#4,

which is clearly not true if AhM < 1, i.e. if A < (hM)-‘. Hence the result. If f(u) = exp(az4/(a + u)), f is convex for 0 6 u < (1/2)(u( cx- 2) and is concave for u > (1/2)a(cr- 2). Hence f’(u) ~fl((l/2)a(cu2)) for all u 2 0 and so (1.2),, has a unique solution for 0 s A < {[(4/a2) exp(a - 2)] . M}-‘.

Uniqueness results for a class of positone problems

Similar consideration

show that (1.3)* has a unique solution for

0 C A < {[exp(-k/(1 where t = ((k - 2)a-

225

+

t))] . [(k(a-

2)/(k + 2(1 + a)).

3. CASE

WHEN

t) - (1 + t)‘)/(l

+ t)*]M}-l

n

D = (0,l)

C R

AND 1 IS LARGE

In this section we discuss the uniqueness of nonnegative solutions for large values of A in the case when (1.1)~ is the autonomous ordinary differential equation -u”(X) = A!( U(X))

in (0, I), (3.1)A

U(0) = 0 = U(1). We first collect some known results.

THEOREM3.1. (see [l, 41). Let f satisfy (fl), (f2) and (f3a) or (f3b). Then for any p E (0, r) there exists a unique k > 0 such that (3.1)~ has a nonnegative solution u satisfying ])~11: = SUE u(x) = u(i) = p. Also for p E (0, r) the corresponding A.is given by XE [A(p)]“’

= (2)lD [

(F(p) - F(s)) -l@ ds

(3.2)

= (2)“‘p l1 (F(p) - F(pu)) -I’* dv

(3.3)

where F(U) =_J?jf(t) dt and u is symmetric with respect to x = 1 and is given by

WI u3 = I0‘@)(F(p)

- F(s))-‘/‘ds

on [0, +I.

(3.4)

Further p 4 A(p) is a continuous function for 0 < p < r and A(O)= 0. Also by the use of the Lebesque dominated convergence theorem it follows that $i(n(PV21

=

w2~l (F(P)

- F(P4)

- iP(f(P)

(F(P) - 6P4)3’2

- Uf(P4)

du ’

which we can also write as (3.5) where H(u) := F(U) - (1/2)uf(u). W e now analyse (3.5) to obtain uniqueness results for large A. THEOREM3.2. Suppose f is a non-decreasing

function which satisfies (fl) and (f3a) and that

lim uf’(u) < lim f(u). u--t+CC U” +m Then there exists 2 > 0 such that (3.1)~ has a unique nonnegative

solution for all il > A.

R.

226

Proof.

Since H(u) = F(u) - (f)uf(u) H’(n)

and hence

SHIVAJI

we have = G)(f(u)

- nf’(n))

(3.6)

H’(u) > 0. Thus there exists a p > 0 such that for every p > p, H(p) > H(u)

.lm=

for all 0 d u
it follows from (3.5) that d/dp(h(p)) > 0 for all p > p. (Note that ave the same sign for A > 0.) But lim A(p) = + x (see [4]). d/dp(Q)) and d/dp{(0))“2) h [’_ + % Hence we have the required result. If f(u) = exp(au/(cu + u)), then lim f(u) = exp (Yand lim uf‘(u) = 0. Hence it follows U’tX u-t= for the ODE case that (1.2)1, has a unique nonnegative solution for sufficiently large /1. n THEOREM 3.3. that (3.1)~ has Proof. Since f(r) = 0, H(Y) Hence

Then

Suppose f satisfies (fl), (f3b) and a unique nonnegative solution for f’(r) < 0, there exists a c’ E (0, = F(r) > F(u) > H(u) for all u E

that f’(r) < 0. Then there exists x > 0 such all A > 2. r) such that f’(u) < 0 in [c’, Y]. Also since (0, Y), and in particular H(r) >py H(u).

there exists a rl E (c’, r) such that H(Y~) >p{

from (3.6) we have H’(u) > 0 in and so from (3.5) it follows that the required result follows. It is easy to see that theorem negative solution for sufficiently

H(u).

Further,

sincef’(u)

< 0% [c’; r],

[c’, r]. Hence whelp E [r], r) H(p) > H(u) for all u E (O,p), d/dp{A(p)} > 0 in [rl, r). But lim A(p) =+ 00 (see [4]). Hence p+ T3.3 implies for the ODE large A. n

case that (1.3)~. has a unique

non-

4. CASE WHEN D = D,; n a 2 THE OPEN UNIT BALL IN iw” In this section we discuss the uniqueness of nonnegative the case when (l.l)* is the partial differential equation -Au(x)

is strictly

increasing

of A in

for x E JD,

where D,; n 2 2 is the open unit ball in [w” andf useful results. Recall us (see 2.3).

f

for large values

forx E D,,

= Af(u(x))

u(x) = 0

LEMMA 4.1. Suppose &, satisfy

solutions

satisfies

and satisfies

(4.1)A

(fl) and (f3a). First we collect some

(fl) and (f3a). Then for (l.l)~, UJ,and

(i) uh(x) 2 ilf(0)uO(x) in D and (ii) if uh(x) + CIA(x), then us < UJ,(X) in D. Proof. Follows

from the maximum

principle.

n

LEMMA 4.2. (see [5]). Suppose f is strictly increasing nonnegative solution u of (4.1)~ is radially symmetric

and satisfies (fl) and (f3a). Then and Ju/dp < 0 in 0 < p < 1.

LEMMA 4.3. Suppose f is strictly increasing and satisfies (fl) and (f3a). Then WJ,= tin - ul musi attain its maximum at the origin, i.e. x~, must be the origin.

any

for (4.1)~,,

221

Uniqueness results for a class of positone problems

Proof.

By lemma

4.2, (4.1)~ reduces d ---1 Pldp p n-1; i P

to

G(P)))

forp

U(P) = 0 and so WA(P) = z&(p) - us --

= 1

satisfies = --$(wA(P))

“-1~(WA(P)))

-y$$J*(,))

= A{f(h(P)) W(P) = 0 the case WA(P) f 0. Then

We shall consider

for0 Sp Cl.

= hf@(P))

M(P)

= h(P)

for0
- fMP)N forp

(4.2)

= 1.

by lemma

- Q(P) > 0

< 1,

4.1 (ii) forO6p


forOsp

Cl.

and so Vf @A(P)) - f h(P))1 Now by lemma

> 0

4.2 ($(~A(P))}p=O

= {$(“n(P))}p_,

= 0.

Hence {d/dp(wn(p))}P,o = 0. Also applying L’Hopital’s rule to (n - 1)/p d/dp(wn(p)) in (4.2) we obtain -n{d2/dp2 ~(p)}~=~ > 0. Thus WA(P) has a local maximum at p = 0. Further if for 1 > p. > 0 {d/dp(wh(p))}, =P0 = 0 then (4.2) gives -{d*/dp*(wh(p))}, =Po > 0. Hence for 0 < p < 1, w*(p) has no local minima. So WA(P) has a single local maximum and thus WA(P) attains its maximum at p = 0. n LEMMA 4.4 (see [6]). Let

subject

to Dirichlet

G(x, y) be the Green’s boundary conditions. Then G(O,y)=KZlog

{

&

function

I

= 4+where

K2, K, are positive

Let f be a strictly increasing exists a constant 2 > 0 such that

there

Proof.

function

G(1

satisfying

that (4.3) implies

that f satisfies

(fl).

Suppose

in addition

nonnegative

that

(4.3)

for all u 2 0.

exists /! > 0 such that (4.1)~ has a unique

Note

;n=2

l):n>2

Z f’(u) Then

-A

constants.

THEOREM 4.1.

there

on D = D, for the operator

solution

for all I. > A.

(f3a). Now since f’ 2 0, from (2.2) we have

228

R. SHIVAJI

w(x$ s

G(xA,Y)~'(@I(Y))~Y.

h(x&fD

S(A):= A

Hence

I

G(~A,Y)~'(&(Y))

D

We shall show that (2.2) is violated by showing that

dy 2 1. lim S(A) = 0. Now by lemma 4.3 ,%++a

S(A)

=ASD,,G(O,

y)f’(W))Q

,

and

so

W) c A

6.G(O,Y>

by lemma 4.1(i)

But since f&(y) 5 us, S(A)

c

AZ

@I(Y) 3

Wo)uo(Y)

&

0,. Hence

in

dY (1 + MO)uo(Y))2

1

I

= AZC,

dy.

1

Y)

G(O,

I

’

(1+ MY>>'

P)

w,

0

1

(P”-’ dp)

(1 + f!f(Oo(P)Y

where C, is a positive constant. Here u. satisfies n-1;

= lforOSp
(Uo(P)))

=Oforp

=1

and hence uo(p) = & (1 - p*) for0 Gp S 1.

Hence, also using lemma 4.4, we have 1 1+ y i

,(p

dp) whenn

(1 - p’))

and S(A)

1

s AZC,K, 1

n = 2. Let q = -_!-

and

f(O)

I1(lodlh4)

A(4 : = A

1 2

l+;(l-p*)

!

(P dP).

=2

Uniqueness

results

for a class of positone

problems

229

Then A(k) = -tq*

c tq2

1’2p(logp) dp (1 -py

= IZl(t)l + lZdt)/(say).

Clearly lim Zl(t) = 0. Now 1-0

Z2(t)

=

-

tg2h?P)

= -$%3P) =

Clearly !$ II(t)

$

(tq + ; _ p2) dP

l Ij +$jT:p(tq (tq + 1 - p2) l/2

+‘, _pi)

dP

Jl(t) + J2(t) (say>.

= ky $

1

(log 4)

(tq + (3/4)) = O.

Also J2Q)

c Q2 j--2 ctq + : _ p2) dp

1 (tq + 1)“2 +p = tq22(tq + 1)“’ i log(tq + 1p -p

1 )I

l/2’

so iii- J2(t) = k;

(tq + 1)“’ + 1 $ (log (tq + 1p - 1) 2

=

liil+$

(

log

tq + 2 + 2(tq + 1)“2

tq

1

and hence, when n = 2, lim S(h) = 0. Now we consider the case n > 2. Let q = - 2n and k++CC f(0)

230

R.

SHIVAJI

Then B(A)

Clearly

lim &(t) r-0

=

tq*

=

L(t) + L2(0 (say)

= 0. Now qf)

<

(12- 2)q2tl 2

1

dp

I 0 (tq + 1 -p*>

1 (tq + 1)“2 +p =--(n - 2)q% 2 2(tq + 1)“’ i log (rq + 1)“2 - p 1

I

0'

so

and hence,

when II > 2,

lim S(A) = 0. Hence the result. I+-s If f(u) = exp(au/(a+ u)) then f(u) d exp(&) for all u 2 0 and when LYY,4, (l/ (a+ U)“) G (l/(1 + U)‘) f or all u 2 0. Hence f’(u) 4 (exp(a))(Y*(l/(l + u)‘) for all u 2 0 and so for the case when D = D, (1.2)~ has a unique nonnegative solution for sufficiently large A. The corresponding result in the case when f(3b) is satisfied [e.g. (1.3)1,] has been proved by Dancer in [3] using degree theory arguments. For large h it seems reasonable to conjecture that there exists a unique nonnegative solution in the case when D is any region in R”. However it seems to be much harder to prove and it would be of interest to produce techniques to deal with it. Acknowledgement-It

is a pleasure

to thank

Dr K. J. Brown

for many valuable

discussions

on this problem

REFERENCES 1. BROWN K. J., IBRAHIM M. M. A. & SHIVAJI R., S-shaped bifurcation curves, Nonlinear Analysis 5(5), 475-486 (1981). 2. AMANN H. & HESS P., A multiplicity result for a class of elliptic boundary value problems, Proc. R. Sot. Edin. 84A, 145-151 (1979). 3. DANCER E. N., On the structure of the solutions of an equation in catalysis theory when a parameter IS large, J. di#. Eqns 37, 404-437 (1980). 4. LAETSCH T. W., The number of solutions of a nonlinear two point boundary value problem, Indiana Liniu. Mark J. 20, 1-13 (1970/71). 5. GIDAS B., WEI-MING NI & NIRENBERC L., Symmetry and related properties via the maximum principle, Communs. Math. Phys. 68, 209-243 (1979). 6. FOLLAND G. B., Introduction to Partial Differential Equations. Princeton University Press (1976).