Pairs of positive solutions for the one-dimensional p-Laplacian

Nonha?

Petgamoa

Analys&. Tkmy, Metho& & Application% Vol. 23, No. 5. pp. 669481.1994 Copy&@@ 1994EkvierScienaLtd Fvintedint3leatLtrwn.All~urelavcd 0362-546x/94 57.00+ .a0

PAIRS OF POSITIVE SOLUTIONS FOR THE ONE-DIMENSIONAL p-LAPLACIAN C. DE COSTBR Departementde Mathkmatique,UniversitCCatholiquede Louvain, Chemin du Cyclotron 2, B-1:348Louvain-la-Neuve, Belgium (Received 18 May 1993; received for publication 4 August 1993) Key words and phmes: Positive solutions, plaplacian, degree theory.

multiplicity results, upper and glowersolutions,

1. INTRODUCTION

THIS PAPER deals with the boundary-value

problem

((pp@‘))’ + f(& u) = 0,

(1.1)

u(0) = 0 = u(T),

(1.2)

where (pP(s) = Isl”-“s, p > 1. W e consider the so-called sub-super-linear case, i.e. the case where, loosely speaking, the “slopef(t, u)/(p,(u) is greater than the first eigenvalue near 0 and near infinity”. As is obvious from easy examples, these assumptions alane do not give the existence of positive solutions. To obtain such a result for an elliptic Dirichlet problem, de Figueiredo an Lions [l] suppose, moreover, the existence of a strict upper solution and obtain the existence %f two positive solutions. An alternative assumption is introduced in Correa [2]. Both results can be adapted to problem (l.l)-(1.2). In that case, the condition of Correa implies the existence of a strict upper solution. Also, we can prove that the assumption near infinity implies that we have an a priori bound on the upper solutions of (1.1) satisfying the boundary conditions (1.2). This, together with the existence of a lower and an upper solution, is enough to have th existence of a pair of solutions of (l.l)-(1.2). The idea of this result goes back to ‘Brown an BaBudin [3]. The paper is organized as follows. We will first develop a theory of loweriand upper solutions for the boundary-value problem Mu’))’ + f(& u) = 0,

(1.3)

u(0) = 0 = u(T),

(1.4)

where (p: IR--) IRis an increasing homeomorphism such that q(O) = 0 andf is an L’Caratheodory function. As a first result, we prove the existence of a W’*‘-solution between a lower and an upper solution. Our main result proves that if there exists a “family” of lower and upper solutions and an a priori bound on all the upper solutions of (1.3) which tisfy (1.4), then the problem (1.3)-(1.4) has at least two solutions Ui E W**‘. The ‘arguhent is ased on the LeraySchauder degree theory. Several papers consider the problem of lower d upper solutions in the Caratheodory case when Q)(S)= s. See, for example, Adje [4], Cost and Goncalves [5], + Habets and Sanchez [6], and Nieto [71. Usually the authors restrict the use of Leray-Schauder degree to a modified problem. This puts a stop to a multiplicity analysis. Let us notice also that 669

670

c. Da COSTER

Deuel and Hess [8] consider the problem of lower and upper solutions for a generalization of thep-Laplacian which is in some sense more general than ours but which requires a growth restriction on p. In the third section, we consider the problem (1. 1)-(1.2). The behaviour off near 0 and near infinity is controlled from a growth conditionf(t, U) 2 b(t)qr,(u) and the spectral assumption Ai < 1, where A, is the first eigenvalue of the spectral problem ($$A0

+ ~~U)e$#(~) = 0,

U(0) = 0 = u(T).

The behaviour off near zero implies the existence of a lower solution, and the behaviour near infinity gives an a priori bound on the upper solutions of (1.1) which satisfy (1.2). Lastly, we give some assumptions on f to ensure the existence of an upper solution of (l.lk(1.2). The conditions which are given here improve the corresponding ones of de Figueiredo and Lions [ 1] and Correa [2]. Finally, the results of this third section are improved in the fourth section for the boundaryvalue problem 24”+ f(t, u) = 0,

u(0) = 0 = u(T),

which corresponds to p = 2. In this case we relax the condition at infinity. Before ending this introduction let us fix some notations. We will use the following spaces: Lp = (u: [0, T] + IR measurable ) jzIu(t)lp dt c 00) with norm II&P = (j,‘lu(t>l” dt)l’P for 1 sp<=; L” = {u: [0, T] + I? measurable I u is essentially bounded) with norm

Il4lP =

esssup((u(t)): t E [0, T]l;

WISP = {u: [0, T] + IRabsolutely continuous and such that jlIu’(t)lp dt c a); #*” = (u E W”P I u(0) = 0 = u(T)]; C’(J, K) will denote the space of function f: J + K which are k-times continuously differentiable. In the case where K = IR,we will only write @(J), and in the case where J = [0, T], we will only write (9. For a function f E L’ we will write f = 0 if f(t) = 0 for almost every t E [0, T], and f # 0 in the opposite case. We will use the abbreviation a.e. t for almost every t E 10, T]. 2. GENERAL

RESULTS

In this section we will consider the general case Mu’))

+ f(& u) = 0,

U(0) = 0 = u(T),

(2.1) (2.2)

where (Gl) 8: IR + IRis aqjncreasing homeomorphism with inverse I,Yand such that p(O) = 0; (G2) f: [0, T] x [R + ,k is an L’-Carath&dory function, i.e. (i) for each II o ,lR,f( - , ti) ‘is measurable, (ii) for a.e. t‘tz [0, T],,f(t, -) is continuous, (iii) for every R > 0, there exists a function CR E L’(0, T) such that, for a.e. t E [0, T] and all u E R with lul 5 R, we have If(t, u)l s CR(t).

671

Pairs of positive sohaions Let F: C?,([O,T], W) + L’ be the Niemytski operator correspondingtolf defined by K(h) = u if u is the unique sohrtion of -Mu’))

and K: L’ + C?ebe

= W,

U(0) = 0 = u(T). Then the problem (2.1)-(2.2) can be equivalently written as . u = .KF(u),

where KF: e&O, T], iR) + C?,([O,T], I?) is completely Garcia-Huidobro et al. [9] for a detailed proof). Let us recall the following definition.

continuous

(we ‘refer the reader to

Definition 2.1. A function CYE Wz*r(O, 7’) ‘will be called a lower solution of (2.1)-(2.2) if (i) for a.e. t t5 [0, T], (&d(t)))’ + f(t, u(t)) B 0; (ii) cY(0)4 0, cu(T) .I 0. The function CYwill be called a strict lower solution of (2.1)-(2.2) if all the above inequalities are strict. In the same way we define an upper solution of (2.1)-(2.2) and a strict upper solution of (2.1)-(2.2). THEOREM2.1. Let Q,and f satisfy (Gl) and. (GZ). Assume. there exist a lower solution (I! of (2.1)-(2.2) and an upper solution j3 of (2.1)-(2.2) such that (I! s 8. Then thelproblem (2.1)-(2.2) has at least one solution II with, for all t E [0, T], cY(t) 5 u(t) 5 p(t). The proof of this result is a straightforward extension of the corresponding result for the second-order equation (Q(U) = u). One proves that any solution u of the modified problem (cW))’ + fit, u) = 0, U(0) = 0 = u(T),

(2.3) .

(2.4)

where At, u) = f@, o(0) + a=uM.+)

- U),

if tl <‘a(t), if fd E b(t),

= f(t, a, = At9 B(t)) + =tN/%t)

- 4,

B(t)l,

if tl > p(t),

(2.5)

is such that CYs u s 8. Next, using Schauder’s fixed point theorem, it ib easy to see that

(2.3)-(2.4) has a solution. With the assumptions of this theorem, we cannot exhibit a set 0 such that deg(1 - XT’,SJ) # 0. To obtain such a result, we have to reinforce the assumptions on upper aad lower solutions. 2.1. Let B, and f satisfy (Gl) and (02). Assume there exist YY, fl E W2B’(0,T) such that CYI jYand, for some I > 0 and all e E IO,a, the functions CY- k and /I + E are,

P~oposrrro~

672

C.DECOSTER

respectively, lower and upper solutions of (2.1)-(2.2). Then deg(1 - KF, 0) = 1,

where n = {UE C?([O,T], R): v t E [O, T], o!(t) - f < u(t) < #l(t) + a]. Proof. Let us first prove that any solution 24E fi of (OW))’ + VU, u) + (1 - J)f(C u) = 0, u(0) = 0 = u(T), where f-is defined by (2.5) and 1 E [0, 11, is such that cz I u s j?. Assume by contradiction that, for some t* E IO, T[, we have rnt=(z4 - B)(t) = I@*) - /3(P) = E. Then we have to, t, E [O,T] such that to c t* c tl, u(t) > B(t) on It,,, t,[, u&J = /3(t,,) and u(tl) = f3(tl). It follows that u’(t,,) r g’&,) and for some t2 E It,,, tl[, u’(ta c /3’(tz). Notice that, if s E [to, tl] is fixed, the function B(t) = j?(t) + u(s) - /3(s) is an upper solution. Hence, f (4 B(t)) + (M’(t)))’

5 0

f(4 u(d) + (d/w)))’

5 0.

and, for t = s This gives the contradiction 0 > W&))

- (PUWZ)) - MW,))

- &/Wo))l

41-1MK u(s)) + (1 - A)f 6, W))l - (cp(B’(W] ds s to tz 1 arctan(u(s) - /p(s)) d.9 ;r 0. 2 s to One proves similarly that u(t) r a(t). It follows now from standard argument in degree theory that =

deg(l - KF, Q) = deg(l - KF, 0) = 1,

where P is the Niemytski operator associated with f:

n

Remark 2.1. Notice that if f is continuous and the functions o, j.?are, respectively, strict lower and upper solutions, the assumptions of proposition 2.1 hold. Now we will give a simple application of this theorem in case f(t, 0) L 0. For that, we need the following result. LEMMA2.1. Let q and f satisfy (Gl), (G2) and assume there exists 6 E C?‘([O,T], IR)such that (i) d(t) > 0 for all t E JO,T[, 6’(O) > 0, 6’(T) < 0 and (ii) for a.e. t E [O,T] and all u E (0, S(t)], f(t, u) 1 0. If u is a solution of (2.1)-(2.2) such that u(t) r 0 on [0, T] and u # 0, then u(t) > 0 on IO, T[, u’(0) > 0 and u’(T) < 0.

673

Pairs of positive solutions

Proof.Suppose there exists to E [0, T] such that u(t,) = 0 and u’(t,) = O..It follows from (i) that either (a) there exists tr > r, such that u(t) I 6(f) on [to, tr] and #‘(jr) > 0, or (b) there exists tz < Q, such that u(t) 5 d(t) on [tz, to] and u’&) < 0. In the first case we have the contradiction 0 < q(u’(t,)) - (p(u’(tiJ) = The second case is similar.

s 0. f’(&4Y8)))’ ds = - t1f(s,u(s))ds s ‘0 s4

n

Remark 2.2.(i) can be replaced by s(t) > 0 for all t E [0, T]. PROPOSITION 2.2.Let (p and f satisfy (Gl) and (GZ). If, moreover, f(-,0) a 0 and there exists

an upper solution j? of (2.1)-(2.2) such that B(t) > 0 on IO, T[, then the problem (2.1)-(2.2) has a solution 242 0. If, moreover, f(-,0)# 0 and there exists 6 E @(to, T], R) with (i) a(t) > 0 on IO, T[, 6’(O) > 0, 6’(T) < 0 and (ii) for a.e. t E [0,T] and all u E [0, s(t)], f (t, u) 1 0, then this solution is such that u(t) > 0 on IO, T[,

U’(0) > 0,

u’(T) < 0.

Proof.The first part of this result can be deduced from theorem 2.1. with (Y= 0 and the second part is an application of lemma 2.1. n Now we will prove that if we add to the assumptions of proposition 2.1 the fact that we have an a priori bound on the upper solutionsof (2.1) which satisfy (2.2), then we have at least two solutions of (2.1)-(2.2). The idea of this result follows from Brown and Bu+n [3], who consider an elliptic PDE and prove that, under such an a priori bound and the fact that f (a,0) > 0, f(*,r)I 0 for some r > 0, there are at least two solutions. We need the following result. LEMMA2.2. Let u, denote the solution of

(CPW)’+ a = 0, u,(O) = 0 = u,(T). Then we have liml,,~,.,,~~u,~~~ = +m. Proof. Let tM be such that l]u,,ll_ = lu,&,J.

and then we have, for all u,

We compute

674

c. Da COSTER

The result follows from the fact that

THEOREM2.2. Under the hypotheses of proposition 2.1, if, moreover, there exists R > 0 such that for all a 2 0 (or for all 4 I 0) and for all solution u of (W

))’ + f(C 10 + a = 0,

u(0) = 0 = u(T),

(2.6)

we have /lull- < 4 then the problem (2.1)-(2.2) has at least two solutions. Proof. The result will follow from the excision property of the degree if we prove that deg(1 - XF, B(0, R)) = 0.

Let us assume that, for all a r 0, the solution of (2.6), i.e. the solution tl of u - KFu = Ka, is such that Ilullo. < R. As KF maps bounded sets into bounded sets, there exists R, > 0 such that, for all u E B(0, R), lb - XFull.. I R,, and, by lemma 2.2, there exists d such that ]lKiiljm> RI. Then the problem u - KFu = KC has no solution and deg(l - KF, B(0, R)) = deg(1 - KF - KG, B(0, R)) = 0.

n

COROLLAICY 2.1. Under the assumptions of proposition 2.1 and if there exists R > 0 such that for all upper solutions j? with /3(O) = 0 = /3(T), we have ll/3110. c R, then the problem (2.1)-(2.2) has at least two solutions. Proof. In particular we have that, for all a r 0, all solution n of (2.6) satisfies llullB < R. Remark 2.3. A similar result holds if there exists a bound on lower solutions. 3. APPLICATIONS

TO THE p-LAPLACIAN

In this section, we consider the particular case (PJU’))’ + f(h u) = 0, u(0) = 0 = u(T), where qP: IR + IRis defined, for p > 1, by c&(s) = Islp-*s.

n

IA b l L’(0, T), b(r) z 0 wieh mesIr E (0; T]:: &((I)’> 0) z 0. we denote by A,(b. 7) ehe eigmvdw

of the SpcCtd

first

probkm (+(u’ ))’ + Ab(r k&4) = 0,

(3. I)

u(0) fi 0 * MT).

43.2)

1’1is known that, the corresponding

CigdUncliOn

cv is

of

COn!Unt

Sign

ad

?;o give precise conditions under which we have an upper solution, we wih use the time‘ch’ and’ Lanolin [ 11 I’]‘. mapping as introduced by MLet x E e’(R*, R' ) and consider the equasion~ (*Ju’

))’

t g(u) L- 0’.

(3.5)’

Associated to this equation kt us consider tCe time-mapping function’

where G(X) = j;g(cs, ds and’ 11/p + l’/q = 11.This functiorr T,(x) gives the t,imc between two zeros of a solut,ion of (3.5) with muimum equal to x (see [1ll1]1). For convenience, we state here she assumptions we shah use on 1. (H’t) There exist b > 0, b. E to@; T) such’ that, for ah u E )o, 41 and a.e. I E 10. T1. where b. 2 0 and1 (H2) There exists 8 E C?([O,/I)‘. @‘) such’ that (i’) for al); u E IO. RJ and1 a,.e. I E 10. T]l. /(I, u) s g(u); (ii’) 7;(R) z 7. (Hl2’) There exists j3 E W’*‘(O~,T) such’ tshat for some 1 > B + e are upper solutions of (1i.W(11.2)‘. fir) > 0 on’ ttl. r[ then /3’(O)’> 0 (resp. /J’(T) < 0). (f;(r) lhe exits g E t??t(O;R1, R’) and1t > 0 such’ that, (i) for ah s e IO, R]I and (1. u) E 10, T) x b. J + e]~./(I, (ii) 7,(R) L T. 043) There exist p > 0, b, E C(O, T) such’ t,hat. for alI u

0 and1 ah t E )o, 0 the functions and i’f j9(0) * 0 (resp. B(T) I 0)

u) 5 g(r). 2 p and’ a.~. I E 10. T]‘.

I((. 4 2 b&k,@). where b, $0 and R,(b,, T)’ c 1. Remurk 3.11.Assumption (HIIN)is true itf, for exampk. there exist r, > 0 and bz E f?(O~,T)‘such that: (i) tim8inf, .,/(r, u)/@#) z f&r) r ~0 uniformly in f; (ii) A,(b;,T) < 1.

676

C.

DECOSTBR

Remark 3.2. Assumption (H3) is satisfied if, for example, there exist E > 0 and &, E P(O, T) such that (i) lim inf U_ +J(f, u)/q+(,,(u)2 bZ(t) 2 E uniformly in t; (ii) d,(b$, T) < 1. LEMMA3.1. Assume thatf satisfies (G2) and (HI). Then there exists QIE W’*‘(O, T) such that, for all 1 E 10, T[, cu(i) > 0 and for all ~1E IO, 11, (P&&M0))

+ f(& PLQ(0) 2 0,

(3.3)

o(O) = 0 = Ix(T). Proof. Let cr(t) be the first eigenfunction know that a(t) > 0 on IO, T[ and

(3.4)

of (3.1)-(3.2) such that rnax, Ero,n o(t) = 6. We

(PJD(M(0))’ + J-09 Par(Q) 2 (V)p(I.Mf)))’ + M&j&M)) 1 (I - &(&, T))M)V&M)) n

20.

LEMMA3.2. Assume that f satisfies (G2) and (H2). Then there exists /3 E W2*l(0,T) such that (9.@‘(0))’

B(O)1 0,

+ f(4 #w 5 0,

w7

2 0,

with, for all t E JO,Tr, /3(t) > 0 and if B(O) = 0 (resp. j(T) B’(T) < 0).

= 0) then p’(O) > 0 (resp.

Proof. By assumption, we have to 5 0 and t, 2 T such that the problem (P#‘))’

+ &)

= 0,

u&J = 0 = u(t,),

has a solution /3 with p(t) > 0 on It,,, tr[, /3’(t,) > 0, tY(tJ < 0. This function is the desired one. n LEMMA3.3. Assume thatf satisfies (G2) and (H3). Then there exists R > 0 such that, for every u E W2*‘(0,T) with u(t) 2 0 on [O, T] and (P&‘(t)))’

+ f(h u(t)) s 09

U(0) = 0 = u(T),

(3.6) (3.7)

we have ll&. < R. Proof. Suppose that, for all n, we have a solution u,, of (3.6)-(3.7) such that u,,(t) 2 0 on 10,T] and, for some &, max U, = u,,(&) = n. For n > p there exist t,,, and t,,2 such that for i = L,2, u,(&,,J = P, u,(t) 2 p on h,,l,

tn,21, uN,,,,)

2 0 and u;(tn,2) 5 0.

677

Pairs of positive solutions

Let us prove that for i = 1,2, lUL(tn,i)l + 00 when n + ao. In fact, as on [&i, tn,J (q+;(t)))’ s 0, we have, for i = 1,2, tn

tl-

Is- I

u,(Q) - p =

p =

uXs)dS -(

l”Xtn,i)lT

hi

and then lUA(tm,i)l+ a. For n large enough we have, for all t such that I(,(+$ I p on [t, tJ, Cp,@XO) 2 ~&4(~*,,))

It

-

II&~ > 0.

(3.8)

follows that u,(t) s p on [0, t,,i] and P =

b.1 I0

r&) h 2 ~;-(~JM”J))

-

l1qJL’k.l~’

from which we can deduce that t,,l --) 0. In the same way we prove that {a,2 + T. Now by continuity and monotonicity of eigenvalues we can pick [ti , t2] c IO,T[ such that, for some peelE ]A,@,, T), l[ and tll with q(t) > 0 on ]ti, f2[, we have @M(0))

+ u&J

Pl =

W)Pp(~l(0) 0

=

=

0,

tl&).

(3.9)

(3.10)

For n large enough t,,, 1 < tl , tn,z > tz and u,(t) > p on I = [ti, i2]. Then, on 1, u,, satisfies (Cp,(W)))’ + gW@$&M)) = 0, with s(t) 2 (f(t, u,(t))/q+,(t))) 1 L(t) $ pl b,(t). By Sturm’s comparison theorem (see Del Pino et al. (121) we have to E ]ti, tz[ such that u,(t,) = 0. This contradicts the fact that u,(t) r p on Iti , hl. n Now by simple applications of the previous results we obtain the following theorems. THEOREM3.1. Assume thatf satisfies (G2), (Hl) and (H2). Then the problem (1.1)-(1.2) has at least one solution u with 0 c u(t) on IO, T[. Proof. This result is a simple application of theorem 2.1, and lemmas 3,l and 3.2.

n

Remark 3.3. It is easy to see that iff E C!([O,TJ x lR+, IR)and (i) there exists 6 > 0 such that, for all u E [0, 8j and all t E [0, T], f(t, u) 2 0, (ii) f(-, 0) Z 0, then hypothesis (Hl) is satisfied. THEOREM 3.2. Assume that f satisfies (G2), (HI), (H2’) and (H3). Then the problem (l.l)-(1.2) has at least two solutions q, u2 such that, for i = 1,2, 0 < Ui(t) on IO, T[. Proof. By lemma 3.1 we can can fiid 6 = ,WYE W2~i(0, T) which satisfies (3.3)-(3.4) and, for all t E JO, T[, 0 c G(t) I p(t).

C.DECOSTER

678

Consider the function if 242 G(t),

fit, u) = f(& u), = f(G 40)

if u c G(t),

- (u - W),

and the problem ($$W))’ + fit, u) = 0,

(3.11)

u(0) = 0 = u(T).

(3.12)

As in proposition 2.1 we see that if u is an upper solution of (3.1 l)-(3.12) then u(t) r G(t) on [0, T]. This together with lemma 3.3 imply that the assumptions of corollary 2.1 are satisfied. Hence, the problem (3.1 l)-(3.12) has at least two solutions Ui. These are such that Ui 2 6, and, henceforth, are solutions of (1 .l)-(1.2). n

Remark 3.4. This result is related to the one of de Figueiredo and Lions [l]. They prove the existence of two positive’solutions

of in Q,

-Au =f(u),

on X2,

u= 0, > i, , lim inf,,+,

in a case where lim inf,,,f(u)/u solution.

f(u)/u

> II and there exists a strict upper

THEOREM3.3. Assume thatf satisfies (G2), (Hl), (H2”) and (H3). Then the problem (l.l)-(1.2) has at least two solutions ul, u2 such that, for i = 1,2, 0 < ui(t) on IO, T[.

Proof. This is a simple application of theorem 3.2 and lemma 3.2.

n

COROLLARY 3.1. Assume that (i) there exist .eo > 0 and bt E L-(0, T) such that L,(b,*, T) < 1 and lim$f

$$-$

2 b:(t)

2 e.

uniformly in t;

’

P

(ii) there exist e1 > 0, R > 0, D > 0 such that, for a.e. t and all s E [0, R + EJ, we have f(t, S) zs D with (3.13) where l/p + l/q = 1; (iii) there exist e2 > 0, &$ E L”(0, T) such that I,(&$, T) < 1 and li$IrIf $$$

2 b$(,(t) 2 E2

uniformly in f.

P

Then the problem (1.1)-(1.2) has at least two solutions ul, u2 such that, for i = 1,2,0 on IO, T[.

c Ui(t)

679

Pairs of positive solutions

Proof. This result follows from remarks 3.1 and 3.2 and theorem 3.3 with g = D.

n

Remark 3.5. (a) This result has to be compared with the one of Correa [2]. He proves the existence of two positive solutions of in SI,

-Au = f(x, u), u = 0,

on

an,

where b,‘(x) = lim, +,,+f(x, u)/u, b:(x) = limU+oof(x, u)/u are such that &(b,*) < 1 and &(bz) < 1. Further, f(x, U) has to satisfy a third condition which in case /$I = [0, T] implies that f(t, U) < uO/C on [0, T] x [0, ZQ,],where C = er - 1. Such a conditioh implies (3.13). (b) By putting together the ideas of this paper and of the paper of Manasevich et al. [13], it is straightforward to improve the result of Kaper et al. [14]. Along the same line, it is also an exercise to prove existence of solutions of (l.l)-(1.2) using better conditions than the one considered for an elliptic Dirichlet problem by Ambrosetti and Rabinowitz [15], Brezis and Turner [la], de Figueiredo [17] and for a second-order ordinary case by Njoku and Zanolin

WI. 4. THE CASE p = 2

Consider the problem U” + f(t, u) = 0,

(4.1)

u(0) = 0 = u(T).

(4.2)

We can weaken (H3) into (H3’) (i) (ii) (iii)

there exist p > 0, b, E L”(0, T) such that b, $ 0 and for all u B p and a.e. t E [0, T], f(t, u) r b,(t)u; Iz,(b,, T) 5 1; lim inf,, +_ If@, 10 - hou)4 $ 0.

LEMMA4.1. Assume that f satisfies (G2) and (H3’). Then there exists R > 0 such that, for all a r 0 and all u E W’*‘(O, T) with u(t) 1 0 on [0, T] and

u”(t) + f(t, u(t)) + a = 0,

(4.3)

u(0) = 0 = u(T),

(4.4)

we have Ilu!o. < R. Proof. Let us define the function g(t, u) = f(t, u) - wL¶9 v&o(t)u and observe that (a) for all u B p and a.e. t E [0, T], g(t, u) r 0; (b) lim inf,, +_ g(t, u) $ 0. Suppose that, for all n, we have a solution U, of (4.3)-(4.4) such thq max u, = n. This function can be decomposed into the form

u,(t) = wl(t)

+ h(t),

c. h &.WBR

680

where e, denotes the eigenfunction jref(t)dt = 1 and

corresponding

T

to the first eigenvalue A,@.,,, T) such that T

w;(t)e;(t) dt

= 0 =

s0

Utk(t)e,(t)

dt.

s0

We will first prove that (w,) is a priori bounded. By a result of Mawhin [19] we know that there exists A L 0 such that, for all n,

Ilw,llm5 A I

A

sok + s

W,,

Tk(t, u,(t))

T)UtMt)le,(t) dt

s T

+ a +

c,(t)]e,(t) dt + A

c,(t)edt)dt,

(4.5)

0

0

where cP E L’(0, T) is such that, for a.e. t E [0, T] and all u E R, g(t, u) = -c,(t).

(4.6)

If we multiply (4.3) by e, and integrate, we obtain T k(t, u,(t)) +

s0

4edt) dt = 0,

(4.7)

which, together with (4.5), gives

Ilw,llo. 5 21\.Tc,WdtJdt s0 and w, is bounded. So if IIu,&, + 00, it means that a, + 00 and u,(t) + COfor all t E IO,T[. By (4.6) we can apply the Fatou lemma on (4.7) and we obtain the contradiction

s

T

T

0 = lim [g(t, u,(t)) + n-r- 0 where we have used (b).

lim inf [g(t, u,(t)) + ale,(t) dt > 0,

4edO dt = s0

n-+ca

n

THEOREM4.1. Assume that f satisfies (G2), (Hl), (H2’) or (H2”) withp = 2 and (H3’). Then the problem (4.1)-(4.2) has at least two solutions ul, u2 such that for i = 1,2, 0 < q(t) on

IO,T[. The proof is similar to that of theorem 3.2. Acknowledgements-The author wishes to thank very heartily P. Habets and F. Zanolin for many discussions and more particularly F. Zanolin for suggesting the problem. REFERENCES 1. FIOUEJRE~D. G. DE & LioNs P. L., On pairs of positive solutions for a class of semilinear elliptic problems, Indiana Univ. math. J. 34, 591-606 (1985). 2. CORREA F. J. S. A., On pairs of positive solutions for a class of sub-superlinear elliptic problems, Dlff. Integral Eqns 5, 387-392 (1992).

Pairs of positive solutions

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