Eigenvalue problem for a class of cyclically maximal monotone operators

E6

ayl *d, JO) d, 30 uoylmryxolddtz vpiso~ aql~o3 uIa[qoJd an[w.ta2?!a aql ‘s3oold aql %uyulamo3 wolelado Drjpads 30 Alp~uolouor,u [vw~xew aql Suqsqqvlsa sl[nsal u~ouy %u!sn Aq pau!elqo aq um suogmgdde laql.xnd .uopenba [e!lua’a33Fp [c!l~l?d Dydl[[a Jeauquou Japlo -puoDas I?01 uopmqddr? a[dwls I? tji!~ apnpuo3 aM ‘d u: asuap s! (h)g pm d au03 F?~I!M padd!nba so aceds llaq[!H aql uaqM suo!ln[os aAg!sod 30 amalslxa aq) aaoId 01 pasn so anbruqDa$ azues aql wop!puo~ AlyplaoD E sagsyes h l~ql amnsse aM ‘H u! asuap s! (h)g leql %uplnbal30 pealsu! lnq paAo[dura sy r.ua[qold lsrg aql u! SC d, uo uoypuo~ ssaulDvdmo3 arms aql asm s!ql UI .paaoJd s! (9

‘a* c (4 r -(he)

0 < X = lo/

JO suoIln[os 30 amals!xa aql alaqM ‘wa[qold G,[mp,, e lap!suoD os[r! aa ‘suopn[os 30 sIred lm!ls~p dtmu A[al!uguy sassassod (r) uaql ‘pagsys ST‘0 z y [[e ~03 lDedcuo3 si { y 5 (x)d, Iff 3 x} : uoyllpuo~ aql pm uarla SF h ‘H u! asuap s! {co > (x)h 1~ 3 x} = :(h)a ‘d, 30 u!ewop aAyaJa aql31 leql allold aM ‘as??3 mo UI ‘suoyn[os 30 sI!ed lsuys!p kmu h[al~ugu~ seq ‘8 = 1n 1‘nd = (n),h uogenba aql uaql (amanbasqns lua%aauo~ B sassassod ‘n it? ch 30 aAy?a -pap laqD?Jd aql a$ouap (n),h alaqM ‘0 01 satham “n (“n ‘(%),h) - (“n),cb put2 papunoq si ((“n)d, I qs!qM BUO[E ‘8 = 1% I qlrM ‘(“n) amanbas AlaAa lvql qms 0 < x we s! alay .a.!) alaqds awes .103 uo~l~puo3 a[euIS-SpqEd aql sags!les pm ‘Mo[aq ruol3 papunoq ‘uaha SF‘(a ‘f&3 3 d, ‘azteds l.raq[!H [?XIo!suaw!p ai!urJu! [r?al e s! H 3: leqi ([e] 30 o[‘z. t.uaJoaql a[duwxa 103 aas) u~ouy s! 11 (o)d, pm [co ‘01 ui san[m sir sayel d, lEq1 aumsse ue3 aM dlgelaua8 30 sso[ 1noqirM jo)he 3 0 lcql aumsse aM ‘asm .nzauq aql u! sv .paq!.mald s! 1n I ‘n 30 UIJOU aql alaqM ‘0 =

alo3aIaqL

n7i E (n)dq (!) ura[qoId aql qlrM pauJasuo3 art? aM ‘laded srql u[ ‘H I+ z [[I? ‘03 (x - z ‘ic) 7 (x)d, - (z)~,JJ! (x)&j 3 I( ‘a’! ‘[co ‘co - [ t H :d, uogmn3 xamo3 snonuguoy.uas laMo[ e 30 the [tyuala33ipqns aql s! y ivy1 sueaur slql ‘H aDcds llaq[rH E u! y Jolelado auolouow [euqxmu F 30 smlal u[ .luasald sr am)mJls [euoyepa~ R 3! pay[dur!s A[qmaprsuo3 s! sura[qold qans 30 kpnls aql ‘am Jeauq aql u! sv ‘l! 103 swa[qo_td an[ema%ia %uypuodsamoD .xap!suoD 01 [emleu s! l! alo3alaqL .[ [] azds llaq[IH t? u! sdno.+uas uo~l~equo~ .wauquou 30 s.Iolwaua% 30 ssr?[3 aql qlrM pagyuapr aq um l! amls sysd[em [zuoyun3 .mau![uou u! a[o’ luE]lodtu! ue sh[d awds )Jaq[IH e u! sJolmado auolouour [m.u!xEw~o SSV’I33H.L

[euo!~epe~

‘suo!punJ

xamo3

‘slolmado

auolouom

pm!xm

‘ura[qold

an[mua%!a

.nzau![uoN

:~IOM Lax

(9L61 @‘I- 6Z PW’~~Ql) ys.n

‘90Lcg 1~ ‘uos!pem

‘ujsuom~

JO @slahyn

‘laiuaa

qmasag

wyeuraq~e~

&lIOLVtTEIdO BNOLONOiU ?V&WXVPU ATIV31?3A3 d0 SSV’I3 V ‘tIOd JGI180lId BflWANB913

94

PHILIPPECLBMENI

conjugate function of cp) is studied first. Here we can use the techniques developed initially by Lyusternik, Schnirelmann [ 31, and Krasnoselski [4]. Then we get the results for cpby passing to the limit employing closedness properties of maximal monotone operators. (See [l] and the references of [ 11). Finally a lower bound on the number of solutions is obtained by using arguments related to those of [4]. See also the references of [2]. Different resultsconcerningeigenvalue problems for maximal monotone operators can be found in [5], [6] where the existence of solutions is based on a result of Rabinowitz [7] about global branches of solutions emanating from a bifurcation point which generalizes a previous result of Krasnoselski [4]. In our context, it can happen that no bifurcation occurs. The author would like to express his gratitude to Professor P. H. Rabinowitz for his suggestions and helpful assistance. 2. RESIJLTS

Let H be a real infinite dimensional

separable

Hilbert

space, with scalar product

(. , . ) and norm

I . I. THEOREM 1.

Let cp:H -+ [0, m] be convex and even, satisfying:

co(O) = 0, (x E HIV(X) I A} is compact

(1) for all 1 2 0,

D(q) is dense in H. Then for all R > 0, there exists a sequence

(2) (3)

(j+ u,J E R+ x H, k E N such that:

(a) luk( = R (b) @(u,J SPA’ /+k 2 0 (c) sup &UJ = + 03. ktN

THEOREM 2.

Let cp: H -+ [0, co] be convex and even satisfying: (p(O) = 0,o = @(O). {x E HIV(X) I 2) is compact

for all ;1 2 0,

lim Cpo = *. X IA-* xtD(q,) I I

Then for all R > 0, there exists a sequence

(1) (2) (3)

(v,, UJ E R+ x H, k E N such that:

(b) 8~ - ‘(v,) 3 vkuk, vk > 0. (c) j$f CP*(U,)= 0, where q*(x): =su {(x, y) - (p(y)} is the conjugate function of cp. YJ Let P c H be a closed, convex cone. (In particular if P n -P = (O}, P is a cone of “positive” functions).

Eigenvalue problem for a class of cyclically maximal monotone operators

95

THEOREM 3. Let 40: H -+ [0, co] be convex and satisfy:

(1)

cp(0) = 0, ix E HIV(X) c A> is compact D(q) = P, where D(q) denotes Then for all R > 0, there exists (/1, u) E Rf

x P

(4

for all 1 2 0, the closure of D(cp).

(3)

such that

(a) 1~1= R PUrO (b) %G) 3 /LU (c) V(U) = , i:_fR d4. ” 4. Let cp: H --t R be convex A 2 0. u E H such that (a) juj = R (b) %4u) 3 J u (4 v(u) = py$&). ”

THEOREM

Remark. In the theorems

and weakly continuous.

1 and 2, the conditions

Then for all R > 0, there exists

sup (p(u,J = co and inf cp*(u,) = 0 imply ksN

that

ktN

the equations ?cp(u) ~/Lu, jut = R and (acp)-‘( u) SW, 1~1= R possess infinitely many pairs of distinct solutions. Clearly by the oddness of arp(iicp- ‘), if (CL,u) is a solution, then (p, -u) is another one. Moreover, in the first case, since uk belongs to the domain of Q, (p(y) < 00, hence the number of distinct solutions is infinite. Similarly, in the second case, cp*(u,) > 0, otherwise, by definition su ((u,, y) - q(y)} = 0, hence q(y) L (u,, y) for all y E H. q(O) = 0 implies that uk E CT@(O) and the YER assumption (1) uk = 0, which is impossible since lukl = R > 0. Therefore the number of distinct solutions is infinite. 3. AUXILIARY

LEMMAS

AND NOTATIONS

Let H a real infinite dimensional separable Hilbert space. For R > 0, B, is the open ball of radius R, B, its closure and BB, its boundary. x,, -+ x denotes a strongly convergent sequence and x,, - x a weakly convergent sequence. For f~ C’(H, R), f’(x) is the Frtchet derivative off at denotes the restriction of f to 3B,. x E dB, is called a critical point of fjFBR if f’(x) x. fl,, - (f’(x), x)Rm2x = 0. a E R is a critical value offI__ if a = f(x) for some critical point x offi?, For R > 0, b E R, if every sequence (x”), xn E 8B,, along which f(x,) + b and /j-(x,,)-(f/(x2, x,,)R-~x~/ + 0 possesses a strongly convergent subsequence, we shall say that flc8, satisfies the Palais-Smale (P-S) condition at b [2]. It is known that the notion of genus is useful for the characterization of critical values. Let us recall some facts about the genus of a set. Let Z(H): = {A C_ H - {O)(A closed, symmetric). For A E C(H), let y(A): =Inf {n E NI there exists g: A -+ R” - (0) which is continuous and odd} with the convention Inf 4 = + co. y(A) is called the genus of A [2], [9]. It follows immediately from the definition that if y(A) = k, B E Z(H) and there exists g: A --) B continuous and odd, then y(B) 2 k. In particular, if A c B, y(A) I y(B). We shall use the fact that if A E C(H), A compact, then y(A) < co, and if A E C(H) is homeomorphic to a k dimensional sphere Sk-’ by an odd homeomorphism, y(A)_ = k [2], [4]. For R > 0, k E N, we define $: = {A E dB,IA compact, symmetric, y(A) 2 k). Clearly yf # 4

96

PHILIPPECLBMEN?

k+ 1 c $. When the context and yR f: H -+ R is weakly continuous,f

is clear we shall omit the subscript R. Finally, we recall that if > 0, f(x) = 0 if and only if x = 0, then c,“: = sup inf ,f(x) i-t,,: xtr satisfies: (i) 0 < cc < cx)and (ii) intca = 0 [2], [4]. We are now ready to state the first lemma. LEMMA 1. Let f~ C’(H, R) with If’(x) - f’(y)1 < Mix - yI for some A4 > 0. Let R > 0 and b = inf f(o). Iv1=R If f is even, for k E N, let b,: = inf supf(x) and ck: = sup inff(x). r& xtr reyt? xslAssume that f lasn satisfies the P-S condition at b (resp. b,, c,), then b (resp. b,, CJ is a critical value offl,,R.

Remark. This lemma can be deduced from the results of [2], [4]. For the sake of completeness, we shall give a direct proof here. Proof of Lemma

1. We consider

the following

du - = -[j’(u) dt

associate

differential

- (f’(u), u)R- ‘u]

equation

:

t>o (3.1)

u(0) = x E H. Since the right h,and side is locally Lipschitz, for x E H, (3.1) possesses a local solution interval [0, t(x)). By taking the scalar product of (3.1) with u, we get, if x E aB,,

$uI = -(f’(u),

u)[l

on some

- R-2j~j2]

i

( lu(0)12 = R2.

By setting u = 1u12we get a linear equation for u, on [0, t(x)). u(t) = R2 is the unique solution, therefore u(t) remains on aB,. But on 8B,, the right hand side is globally Lipschitz and by a standard result we get: (i) Vx E as,, 3!u(t, x) solution of (3.1) with lu(t, x)1’ = R’. (ii) Vt > 0, x + u(t, x) is continuous, and odd if f is even. By taking the scalar product of (3.1) with ti we get: lti12 = -d/dtf(u(t, x)) since (u, ti) = 0. Therefore f(u(t)) is decreasing. Since lu(t)l = R and f maps bounded set into bounded sets, f(u(t)) is bounded. There exists N(X)E R such that: (iii) f(u(t, xl) 1 4~). Consequently (d/dt)f(u(t, x)) -+ 0, as t + co. But (d/dt)(j-(u(t, x)) = -If’(u) -v’(u), u)R-~uI~. Hence (iv) If’(u(t, x)) - (f’(u(t, x)), u(t, x))R-‘u(t, x)1 -+ 0 as t + co. Now consider b (resp. b,, cJ. These quantities are finite since f is bounded on 8B,. We claim that b (resp. b,, ct) is a critical value if for each E > 0, there exists x, E 8B, such that b I f(u(t, x,)) I b + E for all t > 0. Indeed, if it is the case, for each n E N, we can find x, E aB, and tn > 0, such that b < f(u(t,, x,,)) I b + l/n and If’(u(t,, x,)) - (f’(u(t,, x”)), u@,, x,))Re2u(t,, x,J I I/n. Then if y, = u(t,, x,,), y, E dB,, f(y,) + b and If’(~,,j - (f’(y,), y,)R-‘y,J + 0. By (P-S) at b, there exists a subsequence y,, + y and cle a rl y y is a critical point of flasn and b = f(y) is a critical value.

Eigenvalue problem for a class of cyclically maximal monotone

First

consider

the case of b = inf f(u). By definition,

97

operators

for each E > 0, there exists xc E 8Z3, such

lol=R

thatf(x,) I b + s.Butf(u(t, x,)) I f(x,) I b + sforallt > Oandf(u(t,xJ) 2 bsinceu(t,x,)E8BR. This concludes the inf case. Now let b, = inf supf(x). For each E > 0, there exists $ E $ such that sup f(x) I b, + E. xer, TEYf xer Let h,(x): = u(t; x). Since h,:aB, ---f aZ3, is continuous and odd, hl(T,) E $. Therefore for all t > 0, b, d mm, . Thus there exists X(E,t) E I?, such that b, d f(u(t, x(c:, t))). Now choose a sequence

tn T

cc and define xE “: =x(E, t,). Since r, is compact,

XE,“j -+ xc, t,, T CO. We

shall again denote this subsequence

there exists X,E Ic such that by x,, n and tn. For each t > 0, we have

a@, x(a, t,)) -+ ~0, xE). Hence f(u(r, x(s, t,,))) -, f(u(t, x,)). We have b, I f(a(t,, x(s, Q)) I f(u(r, X(E,t,))) for tn 2 1. Therefore b, I f(u(t, x,)) s f(xJ for each t > 0. But xc E Tz thusf(x,) I b, + E. Thus, b, I f(u(t, x,)) I b, + Efor all E > 0, and we are done. The case of ck : = sup inf f(x) follows reyr xer from the preceding one, by observing that -f satisfies the assumptions of the preceding case. Now let us recall some definitions and results about convex functions in a Hilbert space. For references, see for example [I]. Let #:H + [O, 2-j be convex, lower semi-continuous with ~(0) = 0. Then A = 8cp, the subdifferential of cp is # 4 (since 0 E Q(O)), maximal monotone and .Zf : =(I + AA)-’ is a contraction, defined on ~ all H~ for 1 > 0. A,: = A-‘(I - .Z$ is called the Yosida approximation of A. We have D(q) = D(@) and Int D(q) = Int D(acp). cpA(x): > O,?,(O) = Oandcp; = A,. = jGnL((21)- ’ (x - y(’ + q(y)) = iA(A,x(’ + cp(J,x)isC’(H,R),cp, A, is Lipschitz continuous with constant l/A. cpl(x) 1 q(x) as A 10. We shall use the fact that if lim (p(x)/lxl = co, then (acp)-l is defined everywhere and maps bounded sets into bounded sets. 1x1-m -=D(O) The conjugate

function

of cp, V*(X): = su; {(x, y) -

v(y)},

is convex,

lower

semi continuous

and 2 O,cp*(O) = Oand(x, y) I q(x) + cp*(y)by definition. Also(x, y) = cp(x) + (p*(y)iffx&*(y) or y E acp(x), since arp* = (a~))-‘. We have (rp*)* = v and if e(x): =$l*, ‘pl = (Ae + P*)*. Let us prove this last identity. If A is maximal monotone Z = (I + A)-’ + (I + A-‘)-‘. Indeed if x E H, x = ( + q with < = .Z,x and q = ,4,x. [t, r~] E A since A, c AJ,. Therefore [Q l] E A-’ which is maximal monotone. Here 5 E A-‘q and x = r~ + 5 E PZ+ A-$ = (I + A-‘)(q). Therefore n = (I + A-‘)-‘(x), hence x = (I + A)-‘x + (I + A- ‘)- lx. Since acp is maximal monotone, SO is ;lacp and Z = (I + Idq)- 1 + (I + (A&)- ‘)- ‘. But Z - (I + A@)- r = I&), =(I + (Aaq)-l)-l. Hence a’p* = A-‘(I + (lacp)-l)-l = (AZ + (lacp)-lOl)-l = (AZ + acp-‘)-‘. and (AZ + acp- ‘)- 1 = (a[h + q*])- ’ = a[(h + ‘p*)*]. ThereBut AZ + acp-1 = a[& + cp*] fore ‘pA = (,Je + ‘p*)* + C but q,(O) = 0 and (Ae + q*)*(O) = 0, thus qpI = (Le + q*)*. Finally let us recall some property of convergence. If x E D(acp), then (acp),(x) --+ @q)-(x) as IJO (where d@(x) is the element of &p(x) of minimal norm) and I(acp),(x)l I lacp”(x)l. We conclude this section by stating a lemma which will be used later. LEMMA

2. Let I++:H + [0, a] convex, lower semi continuous, with t&O) = 0. Let An JO as n + 00, and let (ccl,. x,“) E R x H such that (1) ul. -+ a; (2) x1” --+ x; (3) 01xE D(+); (4) $;“(xJ = CI~x1 n n. Then (a) a@(x) 3 c(x (b) lim $/,,(x,,) exists and is equal to Ii/(x). *-CO

98

PHILIPPECLBHENT

Proof of Lemma 2. (a) By the monotonicity

(%“X%” - Il/;“(t4J,n

or I+;“, we have: for all

- a) 2 0

u E Q&q.

Hence (ax - (8$)‘(v), x - v) 2 0 for all u E D($G). Since 8$’ is a principal and crx E &j(x). (b) We have

section of a$, x E D(@)

Icl,,,w- I(/&%,,) 2 (Il/;,,(x,,)Tx - X&l = vn. Henceti,,(x) + cnG ti).,(xJ G It/,,(x) - vn.But Il/Jx) t $(x) < m, since x E D(iW, and

sincexi., +

x7

+;,(x,,) =

~I,xI.,

is bounded, 4. PROOF

(a) Critical values For R 2 0, k E N, let yk: = {r G 3R,jr

E”, q, -+

0

as is $,,(x). OF THEOREM

E C(H), I- compact,

1

y(I) 3 k}. b,: = i@h~~~ q(x). We

claim that bk < co, for each k E N, and each R > 0. First observe that for each R > 0, and each n D(q) contains a k dimensional sphere Ik. If not, since x E D(q) n dB,, with R’ > R only if x E D(q) n 8B,, (by the convexity of D(q) and the fact that 0 E D(q)), we would have D(q) E E, where E, is a 1 dimensional subspace with 1 -c k. But this would contradict assumption (3). Now let R > 0, k E N given. We know that for E > 0, there exists a k dimensional sphere IF+’ contained in 8BR+E n D(q). Let E, = span I:+‘. E, is a finite dimensional subspace. Let 4 the restriction of rp on E,. Clearly D(e) 2 Ii+’ and @ is continuous at each point x contained in the interior of Conv IF+& (convex closure) relatively to E,, in particular on I’:: =aB, n Conv rF+‘. r: is a k dimensional sphere, the genus of I,” is k, then I: E yk. Since If is compact and @ is continuous on it, sup @J(X)= sup q(x) < co, therefore b, = inf sup q(x) < co. 0 % b, I b,, 1 xerk xtl-* l-EYk.,tr follows immediately from the definition. For R > 0, k E N, 2 > 0, let bt : = inf sup qA(x). Since ‘pi r cp as II 10, we have b: < b; if k E N, 8B,

rEy

XCI’

6,: = sup bt < co. We claim that sup ik = co. If not, ksN 1.> 0 there exists C > 0, such that b,” < C for all 1 > 0 and kEN. By the definition of hi, there exists rk E yk such that sup q,Jx) 5 C for all k E N; hence sup (p(Jijkx) I C and sup 1/2klA,,,xj2s C p

5

A

and bt i b,, for A > 0. Therefore

xsrr

X6fr

x&r,

2 R - ,/2C l/Jk, i f x E rk. Therefore there exists k, such that for k r k,, x E rk, lJljkxl 2 R/2. Let rk be the image of rk by Jr,,. f, is compact and symmetric since rk is compact and symmetric, and Jljk is continuous and odd. Since 0 4 r, for k 2 k,, fk E C(H). Since Jljk is continuous and odd_, y(I,) 2 ?(I,) 2 k for k 2 k,. Let r: = {x E HIV(X) I C and 1x12 R/2}. r is not empty since IYkc l= for k 2 k,._r is compact by assumption (2), symmetric since q is even and 0 $ i= by definition, therefore y(I) = k, < 00. But, for k = max (k,, k,) + 1, rz c T, hence y@,) 2 k, and @,) 2 k > k,, a contradiction. Thus sup & = co. But IJijkxl = lx -

l/kA,,,x(

hEN

(b) Approximate solutions Let R > 0, k E N. Let n: = R2/2b, if b, > 0, and 2 > 0 arbitrary 0 < A < X, there exists (P,, A’uk, J E R x H such that:

1%AI= R

if b, = 0. We claim that for

(4.1)

Eigenvalue

problem

for a class of cyclically

maximal

‘p;(‘k,I) = pk, d'k. 2 q&‘,

monotone

99

operators

”

pk.A

(4.2) (4.3)

A) = ‘;’

Let 0 < /z < X. Since ppI E Cr(H, R) and p); is Lipschitz continuous, by the Lemma 1 we are done provided that (P&~, satisfies (P-S) at 6:. Let xn E dB, such that cp,(x,) + b,”and Iq;(x,) - (cp;(xJ, x,)R - ‘x,j +O. We have 91(x,,) = Alxn and xX= Jlxn + IA,x,. Therefore (1 - /Z(A,x, x,)R- ‘)A,x, I 1. We can extract a subsequence, still - (Al+ x,)R - ‘JIx, -+ 0. But 0 I A(A,x,, x,)R-’ denoted by x,,, such that I(A,x,,, x,)R-’ + a E [0, 11. We claim that a < 1. Otherwise JAx,, + 0 since ALxn is bounded and (Ayx,, x,)R-’ does not converge to 0. Then J./~[A,x~/~ = $(AIxn, x.) - k(J1x,,, A,x,) + R2/21. Hence by using the lower semi continuity of cp (Assumption 2), 0 = ~(0) < lim cp(J,x,) = hn~ [(P~(x,,) - J/2(Alxnj2] = b,” - R2/21 < b: - R2/2X I 0, a contradiction. Therefore a < 1. Since cp,(x,) = ;1/21A1x,12 + cp(J,x,) is bounded, cp(J,x,) I C and by Assumption (2), Jlx. lies in a compact subset of H. We extract a subsequence still denoted by xn, such that Jlxn + z. Therefore Alx, converges strongly to al- ‘/( 1 - a)z and x, + (1 - a)-‘~. Thus %ldBn satisfies (P-S) at bi. Finally, observe that in (4.2), p k. 1 2 0 follows from the monotony of ‘pi and the fact that q;(O) = 0. (c) Limit procedure Let R > 0, k E N. We claim that there exists In JO and uk E H such that uk, I._ + uk. Let 0 < 1 < ,I. qn(ukJ = b,” < b, < 03. Therefore ‘p(J,u,,,) I b, and 1/2(A,uk,,12 I b,. By assumption (2) there exists a sequence A,10 and uk E H such that JAnuk, 1. -+ u,.ilA,u, I I2 < 2b, implies that = J, uk I + InA, uk.l -+ uk. In particular iuki = R. -+ 0, hence uk.l ‘nAAnUk, 1, We shall prove that 1~~.I,n is bo&bid. We al;eadi know that pk, I, 2 0. Assume that there exists a subsequence An, JO such that Pi, A7,,-+ + co. For nj big enough, pk. I.j > 0. We have for all u E D(t?cp):

by the monotonicity

of A,. But 1

1 ~

AL,,Uk,A.,

=

‘k I 1 "J +

‘k’

~

A A.,ju-+0

kin,

(A,o is bounded

since v E D(&)). Therefore

we get:

~ __ (u,, uk - u) 2 0 for all v E D(atp) = q(cp) = H by Assumption

(3). But uk # 0, a contradiction. We know that ‘k, J,, + UkTI Pk.i., I < C, therefore we can extract a subsequence, still denoted by such that uk A- -+ uk, pk 1 -+ pk 2 0. We can apply Lemma 2, with $ = cp, ul. = ,uk.L ,x1, = u~.~,. Observe that i,u, E D(cp)‘= H. Therefore we have lukl = R, &p(uk) 3 pkuk and rp(u,) = ]\r~~pJu,,,~) = lim bLn = sup bf = ik. But in the first part of the proof, we showed that sup 8, = co. a>0

A,,10

sup

cp(u,)

ksN

= sup

Hence

kEN

8, =

x)

and we are done.

kcN

5. PROOF (a)

OF

THEOREM

We claim that if $: H + [0, UI] satisfies the hypotheses

2

of the Theorem

2, then D($*) = H,

100

PHILIPPE CLEMENT

I/* is weakly continuous, $*(x) = 0 if and only if x = 0, &,9*(x)3 0 iff x = 0 and I,G*is even. 1,4is convex, even, lower semi continuous by Assumption 2, f f co by Assumption (l), and a$* =(a+)-’ is defined everywhere and maps bounded sets into bounded sets by Assumption (3). Then D($*) = H. I,/I*is convex lower semi continuous and by Hahn-Banach, weakly lower semi 7 continuous. We prove that if xn - x, then hm $*(xJ d $*(x). By the surjectivity of a$, there exists y, E D(&G) with x, ~a$(yJ. Since x, is bounded, so is y, and I&J,) < $(y,) + I#J*(x,) =(x9 Y”) g c. By Assumption (2), Y,, lies in a compact subset of H, therefore ,Ilc (Y,, x,, - x) = 0. But 7 $*(x,) d (y,,xn - x) + G*(x), hence hm $*(x,) < e*(x). Therefore II/* is weakly continuous, Clearly $*(O) = 0. Assume t++*(x)= 0. Then (x, y) I It/(y) for all y E H, hence x E a+(O), so x = 0 by Assumption (1). Clearly 8$*(O) 30. Assume a@*(x) 30, then x E a+(O), hence x = 0. Finally, it is clear that II/* is even, since 1//*(-x) = sup {t-x, y) - q(y)} = sup {(x, y) - cp(-y)> = sup -yeIf tcx, Y) - Icl(Y))= ti*(x). Observe that for all il-> 0, IZe+ cpsatisfies the assumptions of the Theorem 2. (e(x): = $x1”). Therefore for all 1 > 0, cp and (cp*), = (Je + (p**)* = (Je + go)* are weakly continuous, LO, =O iff x = 0, L@*(x) and a(q~*)~(x)3 0 iff x = 0, and q*, (v*)~ are even. Therefore, by a result stated in Section 2, ck: = g7~ $II q*(x) and ct: = g?y (q*),(x) satisfy 00 > clr, c,” > 0 and ${ ck = 0. Since c,” (: ck, Ekk:= sup c:, satisfies E, > 0 and ei 2, = 0. A’0

(b) We claim that for any R > 0, k E N and ;I > 0, there exists uk,1 E H, vk,1 > 0 such that

l%,%l =R @*Y;(Uk.I) =

(5.1) Vk.A Ok.A

(cp*), (Ok,1) = c:.

(5.2) (5.3)

First observe that if (cp*);(u,,) = v~,~u~,~ for some v~,~E R, then vk,+ has to be >O. Indeed, vk Ais 2 0 by the monotonicity of (cp*); and the fact that (q:)‘(O) = 0. It 1s > 0, since (q*)‘(x) = 0 iffx =O,ando,, # 0. Now (5.1)-(5.3) will be proved, if (cp*)l(aBRsatisfies the (P-S) condition at ci, by Lemma ’ 1. Let x, E aB,, such that (v*),(x,) + ci > 0 and 1@*);(x,) - [GP*);(x,), xJ R-’ xn ) -P 0. Since x, E aB,, we can extract a subsequence still denoted by x, such that x,, - x E H. We claim that (q*)‘(x,) + (#J’(x) at least, for a subsequence. First observe that (~p*)~= (Ae + q)*, hence (cp*); = a[((~*),] = a[(Re + q)*] = (a[Ae + cp])-’ = (U + ap)-l. Since lim U+12 + co(x) = Go lxi* m I4 ’ .-D(w) (U + &p)- ’ maps bounded sets, into bounded sets. In particular y,,: = (AZ + acp)- 1 x, is bounded.

But xn - JY, E My,), therefore CP(Y,) I UP + cp*& - JY,) = (xn - AY,, Y,,) S (x,,, Y,> I C for some C > 0. By Assumption (2), we can extract subsequences, still denoted by x, and y, such that y, -+ y E H. Now (y, - (AZ + &p)- ’ U,xn - v) 2 0 for al1 u E H, therefore, 0, - (AZi- acp)- ‘u, x - u) 2 0 for all u E H and y = (AI + acp)-’ x by the maximal monotony of (21 + drp)- ‘. So (cp*Y;(x”) -+ (v*)@) and x,, -+ R2[((~*);(x), x)1- ’ CP*);( x 1 since ((q*)‘(x), x) > v*,(x) > 0. Therefore (q*), I PBnsatisfies the (P-S) condition at c,“. (c) We claim that for each R > 0, each li E N, vk.Ris bounded as AJO. We have (cp*); = a(cp*)>.= (&p*), = ((&p-‘), and I(@J); ‘(v,,,)I I ((dq- b)“(v, *)I. But @cp)-’ maps bounded sets into bounded sets, so Iv,,,1 = R-‘I(cp*)‘(u,,,)) < R-‘\(acp-‘)“(u,~,)J < C for some C > 0. So

Eigenvalue

problem

for a class of cyclically

maximal

monotone

101

operators

5 ((cp*)~)*(u,, J + (rp*)& i uk. J = (uk.A’ vk, Auk, 1)I CR’. But ((cp*),)*= AT+ cp, so ‘p(u,, J < CR’. Therefore uk, I lies in a compact set of H. We can extract a subsequence In JO such +vk. Since vkuk E H = D(rp*), by Lemma 2, we get: ]uk( = R, &p*(u,)~ that ‘k.2, +ukandvk, vkuk and cp*(u,) = &~(~*),Ja,~ J = ;t$ c,” = Z,. dq*(v,) = (@-‘(ok) 3 vkuk. Again vk 2 0 and

((cp*)J* bk,J

even > 0, otherwise” uk E C+(O) which is impossible by Assumption Moreover we already know that 2: 2, = 0, so $I; cp*(u,) = 0. 6. PROOF

OF

THEOREM

3

(a) For R > 0, let b: = , ir”=fRq(x). 8B, n II(rp) is not empty, contradicts

Assumption

(1) and the fact that 1uk1 = R.

otherwise

(3). %hus 0 < b < a3. For I > 0, let b”: =

inf

II(cp) s BR, which

q,(x). 0 I bL I b.

(b) Let R > 0,x: = R2/2b if b # 0 and arbitrary positive if b = 0. ?d “, A < X, rp 1 satisfies (P-S) at b, as in the proof of Theorem 1. Then there exists u1 E H, pi 2 0 such tha;(rb,/ = R; (ii) cp;(u,) = piu,; (iii) cpl(u,) = b,. (c) We can apply the same proof as in Theorem 1, to get the existence of a sequence A, 10 such that ul, + u and ,uL,.,-+ p 2 0. Observe that we get (u, u - u) 2 0 for all u E D(acp) = D(cp) = P. Since P is a cone, we can choose u = 2u and we get the same contradiction as earlier which proves the boundedness of p>.,. So uI, --t u, p,,-+ ,u and pu E P since p 2 0 and u E P. Therefore ,uu E D(q) and can invoke Lemma 2, to get iirp(u) 3 /JU and q(u) = !‘$ (PJu,,). But q(u) = & (pAI,(uk,)= ;up, b” I , b = , tn_fRq(u). Since IuI = R, q(u) = b and this “concludes the proof of Theorem 3. V 7. PROOF

OF

THEOREM

4

Let R > 0. Then BR is weakly compact and cpis weakly continuous. Therefore there exists u E BR such that q(u) = ,?a~, q(u). Since BR and cpare convex, u E aZ3, or can be chosen in aB, if there are more than one maGimum. Since D(q) = H, D(&p) = H. Therefore for all z E aB,, 0 2 V(Z) V(U) 2 (y, z - u), for y E @(u). If 0 E acp(u), we are done. If not, let y E acp(u), y # 0. Then (Ry/lyl, z - u) I 0, for all ZE~B,. By taking z = Ry/ly(, we get R2 I (Ry/(yl, u) hence Ry/(y( = u and C@(u) 3 ilu, for some d 2 0. 8. AN

EXAMPLE

Let R E R” a bounded domain with smooth boundary. Let /I c R x R an odd maximal monotone graph with 0 E p(O) and D(b) = R. Let j: R + R the unique convex function such that /3 = 8j andj(0) = 0. LetH

= L2(n)andcp:

H-+

[0, co]definedbycp(u):

= i

grad2udx

L Jn

and j(u) E L’(R), + co otherwise. It is well-known (see for example

+ c j(u)dxifuEW1’2(R) JO

[8]) that cp is convex, even, lower semi continuous,

D(&p) =

6’. 2(Q) n W2* ‘(Cl) n (u E L2(sZ)) B(u) E L’(Q)} and acp(u) = -Au + p(u). Since the injection of W’,2(Cl) into L2(Q). 1s compact, {u E L’(0) I q(u) I c> is compact in L’(R) for all c 2 0. Clearly D(q) is dense in H. Therefore cp satisfies the hypothesis of Theorem 1 and for all R > 0,

there exists infinitely -Au

many distinct

pairs of solutions

]u12 dx = R2,

+ ,8(u) s;lu, sR

of

u E k?“. 2(Q) n W2* 2(Q)

and

p(u) E L2(Q).

(*)

102

PHILIPPE CLEMENT

By a standard regularity result u E C’sa@2) for c( ~10, l[. If j? is univalued u E C’,‘(Q) and therefore u is a classical solution of (*). Clearly 40 satisfies the condition lim lul-m usD(~)

If we assume moreover infinitely

many

f@=

IUI

+co

pairs of solutions

Wzs ‘(Cl) with /I(u) E L’(Q). As an application of Theorem

to C’(R),

.

that 0 = B(O), then 0 = &$$O). By the Theorem

distinct

and belongs

for (*) satisfying

A2

2, we get the existence

jul’dx sR

3, we shall give the following

=R2,

of

u~l;i7’.~(R)n

lemma.

3. Let H a real ordered Hilbert space with a positive cone P satisfying (u, u) 2 0 if u, u E P andsuchthatforallzeH,thereexistsz+,z-EPwithz =z+ -z-,(z+,z-) =O.Letcp:H-+[O, co] convex satisfying LEMMA

(1)

V(O) = 0 {x E HI q(x) _< c] is compact

for c 2 0

(2)

D(9) 2 P

(3)

(Z+Mcp)-‘PGPforall;1>0.

(4)

Then for all R > 0, there exists u E H, 1 > 0 with (a) (u( = R (b) Mu) 3 An (c) u E P. Proof of Lemma 3. Let @: = cp + I, where I,: H --+[0, CD] is defined by Z,(u) = 0 if u E P and + co if u 4 P. By Proposition 4.5 of [l], &$ = acp + aZ, and D(@) = P. Clearly @J(O)= 0 and (?, satisfies Assumption (2) of Theorem 3, so for all R > 0, there exists 1 2 0 and U E IYB, n P with q,(G) = , m=fR@J(U)and @j(u) 3lU. Let w E al,(U) such that Q(U) + w SAG. We have al,(G) = {z E HI (i, u - ii) I 0 for all u E P}. We claim that if z E aZ,(i& z E -P and (z, U) = 0. Indeed, by assumption exists z+ and z- E P such that z = z+ - z- and (z+, z-) = 0. Hence for all c( 2 0 we have MIZ+I2 - (z, U) I 0. This is possible only if z+ = 0. So z E -P. By taking u = $ii, we get $(( -z), -U) 2 0, hence (z, ii) = 0. So we have ii E P and y = -% E P such that (i) ii + &P(U)3 (1 + l)U + Z; (ii) (U. Z) = 0. Let I,+(U)= $Iui” + q(u) - (1 + l)(ii, u). From (i) and (ii) we get $(I?) = $(ii) - (2, U) I Ii/(u) - (2, y) for all u E H. Since ? E P, I,@) I t,b(u) for all u E P. Since u E P, I@) = i$$(u). But !rpf Ii/(u) = YI$ $(u). Indeed, there exists ii E P such that $(I%) < Ii/(u) for all u E H. Such U is unique and defined by U = (I + acp)-’ (A + 1) ii. By Assumption (4) and since (A + 1) U E P, ii E P. Consequently, by the uniqueness, U = r? and U = (I + 8’~))’ (1 + 1) ii or acp(U) 3 ;lU. This concludes the proof of Lemma 3. As an example we consider again the equation -Au

+ p(u) 3 lu,

IuI = R.

uE

w&2

n

$71.2.

(*)

Eigenvalue problem for a class of cyclically maximal monotone operators

103

We already mentioned that Assumptions(l), (2) and (3) of Lemma 3 are satisfied. It is a standard result that the solution of the equation u - IAu, + @(u) 3 f belongs to L: if 1 > 0 and f E Lt. So the lemma can be applied and for all R > 0, (*) possessesa positive solution, which is C’*a(l(2) by a standard regularity argument. Moreover if /3 is univalued and C’, u E C2.a(Q and U(X)> 0 for x E a. REFERENCES 1. BRBZISH., Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilberl. NorthHolland, Amsterdam (1973). 2. RA~IN~W~TZ P. H., Variational Methoak for Nonlinear Eigenvalue Problem (C.I.M.E. 1974). Edizioni Cremonese, Roma (1974). L. A. & SCHNIRELMANN L. G., Topological Methods in the CalcuIw of Variations. Hermann, Paris (1934). 3. LYUSTERNIK M. A., Topological Methodr in the Theory of Nonlinear Integral Equations. Macmillan, New York (1964). 4. KRASNOSELSKI DA VEIGA H., Differentiability and bifurcation points for a class of monotone nonlinear operators (to appear 5. BEZRAO in Annali Mat. pura appl.). 6. DIAS J. P., Variational inequalities and eigenvalue problems for nonlinear maximal monotone operators in a Hilbert space, Am. J. Math. (to appear). I. RABINOWITZ P H., Some global results for nonlinear eigenvalue problems, J. funct. Anulysu 7, 487-513 (1971). methods in Hilbert spaces and some applicatrons to nonlinear parnal differential equations 8. BR~ZISH., Monotomcuy in Cormibutwzs to nonlinear functional analysis (ed. E. H Zarantonello) pp. 101-I 56. Academic Press. New York (1971). 9. COFFMANG. V., A minimum-minimax principle for a class of nonlinear integral equations, J. Analyse Math. 22, 391419 (1969).