- Email: [email protected]
Periodic solutions of a superlinear wave equation
Nonlinear Analysis. Theory. Methods & Applications, Vol. 29, No. 3, pp. 265-282, 1997 @ 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362.546X/97 $17.00+0.00
Pergamon
PII: S0362-546X(96)00045-4
PERIODIC
SOLUTIONS
OF A SUPERLINEAR
WAVE EQUATION*
D I N G Y A N H E N G and LI SHUJIE Institute of Mathematics, Academia Sinica, 100080 Beijing, ER.China
(Received 19 December 1995; receivedfor publication 28 March 1996)
Key words and phrases: Periodic solution, wave equation, minimax method.
1. INTRODUCTION
We consider the existence problem of periodic solutions for the following wave equation { utt - Uxx = bu + g(x, t, u), x E (0, rr), t E R, u(O, t) = 0 = u(rr, t), t E R, u(x, t + 2rr) = u(x, t), x ~ (0, rr), t ~ R.
(W)
Assume that f (x, t, ~) - b~ + g(x, t, ~) satisfies (fl) f E C([0, rr] × R 2, R) is 2rr-periodic in t; (fE) f ( x , t, ~) is strictly monotonically increasing in ~; (f3) there exist ? > 0 and ct 6 (2, co) such that O
f(x,t,s) ds<_f(x,t,~)~,V(x,t,~)witht~l>-_-i,
(f4) g(x, t, ~) = 0(1~1) as ~ - 0 uniformly in (x, t). Let rn = a2/Ot 2 - aElax 2, and o-(n) the set of eigenvalues of [] subject to the periodic and boundary conditions in (W). Set G(x, t, ~) = ~ g ( x , t, s)ds. We shall establish the following results: THEOREM 1.1. Assume that f satisfies (ft)-(f4). Then (W) possesses at least one nontrivial (weak) solution provided moreover that one of the following cases holds: (1) b ~ tr([]) or b = 0, (2) b ~ o-([]) and there is a 6 > 0 such that G(x, t, ~) >-_0 whenever I~l - 6, (3) b ~ o-([]) and there is a 6 > 0 such that G(x, t, ~) <_ 0 whenever [~l -< 6. THEOREM 1.2. Assume f ( x , t, ~) satisfies (fl), (f3), (f4) and (f~) f ( x , t, ~) is nondecreasing in ~. Then (W) possesses at least one nontrivial weak solution provided moreover that one of the following cases occurs: *This research was partially supported by the Chinese National Science Foundation. 265
266
D I N G Y A N I l E N G and LI S H U J I E
(1) b ¢ o-([3) or b = 0; (2) b E if([]) and there is a 6 > 0 such that g(x, t, ~)~ >_ 0 whenever I~l ~ 6; (3) b ~ o-(E3) and there are 6 > 0, 2~1 > 0, he > 0, 2 < p _< o~ such that - h i [~[a _< g(x, t, ~)~ <_ - A21 ~ I", whenever t ~1 -< 6. Note that f2 (or (f~)) and (f4) imply b _> 0. T h e o r e m 1.1 and 1.2 remain valid under the same hypotheses if (W) is replaced by
u, - u~-,. = - ( b u + g(x, t, u) ) T h e o r e m 1.1 and 1.2 generalize Rabinowitz' result [1] and a result of [2]. In [1] Rabinowitz proved that (W) has a nontrivial solution if b >__0 and ~ - g(t, x, ~) is increasing. In [2] Li and Szulkin had proved T h e o r e m 1.1 but under the strongly monotonically increasing condition for f : there exists an t > 0 such that (f(x, t, ~) - f ( x , t, 0 ) ) ( ~ - 0) >- EI~ - r/I ~ for all x, t, ~, rl. When b = 0, T h e o r e m 1.1 has been proved in [1] and T h e o r e m 1.2 has been proved in [2]. In the present paper we only need to deal with the b > 0 case and we do not ask that ~ ~ g(t, x, ~) is increasing. T h e o r e m 1.2 is m o r e delicate because the variational form of (W) does not satisfy (PS)* condition if f satisfies (f2). O u r p r o o f relies on the approximate m e t h o d and a special m i n i m a x m e t h o d under the local linking condition.
2. P R E L I M I N A R I E S
Let Q = (0, rr) x (0, 2rr) and [I ' lip denote the L P ( Q ) - n o r m for p ~ [1, oo]. We consider the Banach space
X=[u=
(2.1)
j~tq~"ukjsinjxeik';u-~J=-ff~j and llullx < °° t k~Z
with the n o r m
Ilullx
Ij 2 - k 2 - bllukjl 2 +
=
j:-
O,b
~. j:-kZ=b
where cx E (2, oo) is the constant of (f3) and z =
~ . Ukj s i n j x e ikt. j2 =k 2
Clearly X is reflexive. Set
Im:jl 2 + Ilzll~
}'
Periodic solutions of a superlinear wave equation
X~ = [ u c X ;
u=
267
Z Ukj sin j x e ikt , jZ-k2>b ~'. Ukj sin j x e ikt , j2-k2
X~ = f u E X ;
u = j2~k2
x° = {u e x; u=
~"
ukj sin j x e ikt ] ,
j2-k2 =b
{u xu: Ukjsinjxek} /'2 = k 2
where it is understood that if b = 0, X° = X ° = {0}. Then we have the direct sum decomposition
X = X~- ~ X ° ~ X f
~N,
(2.2)
and, correspondingly, u = u + + u ° + u- + z for u 6 X. It is known that X~ are compactly embedded in/_Y for p 6 [ 1, oo), and dim X° < ~. In what follows, for convenience, we introduce the following equivalent norm on X: Ilull -- max{llu+llx, Ilu-IIx, Ilu°llx, Ilzll,~}.
(2.3)
For any index set A C N × Z, let F
Z Ukj s i n j x eikt } (j,k)EA
X^=tu~X;u= and define the projector P^ " X - X/, by
P^u =
~"
Ukj sin j x e ikt.
(j,k)EA
In particular, set/x0 = {(j, k) E N x Z; j2 = / f l }. Note that X^ 0 = Na. LEMMA 2.1. Let h(x, t, s) ~ C([0, rr] × R 2, R) be 2rr-periodic in t and satisfy (ht) there are bl, b2 > 0 such that Ih(x, t,s)l <- bllsl + b2lsl '~-l. Suppose that ^ c N × Z \ ^ 0 and u E X satisfies n(P^u) = P^h(x, t, u) in the sense of distributions. Then IIe^ulloo -< b3llullc, + b4llull~ -1 where b3 and b4 depend only on bl and b2.
Proof. The proof is closely that of [3, Proposition 1] or [2, Lemma 3.1]. We omit the details. Suppose that h ~ C([0, rr] × R 2, R), h(x, t, 0) = 0, and satisfies: (h2) h(x, t, s) is strictly monotonically increasing in s; (h3) h(x, t, s) satisfies (f3). Set, following [1], for L > 0, min ( h ( x , t , s + T ) - h ( x , t , r ) )
I'rl_
e~(s)
=
fors>__0
(x,t)~.-Q max(h(x,t,s+r)-h(x,t,T)) ITI-
(x,t)~Q
fors<0
268
D I N G Y A N H E N G and LI SHUJIE
and /2~"(s) = min (tpjL,(s), - tp~(-s) ). Then tp~(s) and U~(s) are strictly monotonically increasing and tp~'(s) _+~,/2to, (s) - ~ as s - ~ .
-
+_~ as s -
LEMMA 2.2. Let h ~ C([0, rr] × R 2, R), h(x, t, 0) = 0. satisfly (h3) and (h2)' h(x, t, s) is nondecreasing in s. Suppose that u = v + : E X with v E L ~ and : E N~ satisfies 22)(/) -< 0,
f (h(x, t, u) t AIzl' Q
V q) c N,~.
(2.4)
For some A _> 0. Then - ~ L °°. Moreover, if h satisfies (h3) and (h2) instead of (h2)', then for each u = v + z c X with IIvlloo _< L, z ~ N~ and satisfying (2.4), we have 1
t2~(511zllool _< 4tlh(v)ll~
(2.5)
where (h(v))(x, t) = h(x, t, v(x, t) ).
Proof. The p r o o f is similar to [1, L e m m a 3.7] or [2. L e m m a 3.2]. In fact, for any M > 0, define a function q : R - R by q(s) = 0 if Isl -< M, q(s) = s - M i f s > M , and q(s) = s + M if s < - M . Recall that each z E N~ can be represented as z ( x , t ) = p ( x + t) - p ( - x + tl - z + - z with fo" p(-r) d T = 0. where p is periodic with 2rr (see [1]). Set q+ = q(z + ) and q - = q ( z - ) . Then q)=q+-q~ N , , q + >_q- i f z _ > 0 a n d q + <_q ifz<_O, a n d f Q Z q+ = fQz+q - = 0 . By(2.4),
~Q(h(x,t,v+-)-h(x,t,v))(q+-q
-) <_ I Q ( h ( x , t , u ) + A[zl ~ 2 z - h ( x , t , v ) ) ( q + - q <_ - f
JQ
)
h(x,t, v)(q + - q - ) <_ IIh(v)llo~ f (/q+l + Iq-I), JQ
(2.6) where (h(v))(x, t) = h(x, t, v(x, t) ). By the superlinear assumption (h3), for any R > 0 there is a constant CR > 0 such that
]h(x,t,S+T)-h(x,t,T)l Let R = we have
if Ivl _
>-CRIsl-1
vll~o. N o t i n g that, by (h2)', the integrand on the left-hand side o f (2.6) is nonnegative,
~o(h(x't'v+z)-h(x't'v))(q+-q-)>-CR~oZ(q*-q-)-fQ >_ ( C R M -
(Iq+l + iq-I)
1) f (Iq+l + Iq-I), 2Q
which, together with (2.6), implies
(CRM-
1)((Iq+l JQ
+ lq-I) -< IIh(v)ll~ f ( I q + l JO
+ Iq-I).
(2.7)
269
Periodic solutions of a superlinear wave equation If z is not essentially bounded, the integral in (2.7) is positive and so
C R M - 1 _< Ilk(v)Iloo for all M > 0 which is a contradiction. The first conclusion o f the l e m m a follows. Suppose h satisfies (h2) instead of (h2)', and u = v + z with Ilvlloo -< L and z ~ Na satisfies (2.4). I f z = 0, (2.5) is clear. Hence assume z ~ 0. For any 6 _> 0, let Q6 = {(x, t) c Q; Iz(x, t)l >6}, Q~ = {(x, t) ~ Q;z(x, t) >_ ~5}, and Q2 = Q6\Q~. By the first conclusion, Ilzlloo < oo. One has
I
Q;
(h(x't'v+z)-h(x't'v))(q+-q-)
>I
Q;
~JL'(Z)z(q+-q-) >'~IL'(6) I z
and I
Q~
(h(x, t, v + z) - h(x, t, v))(q +
I
q-) >
-Ilzll~
Q;
z(q + - q-).
'I'~(-6)llIe; zlo~
z ( q + - q-)"
Therefore,
I
+ - q-)>_
(h(x,t,v+z)-h(x,t,v))(q O~
>_
~11~(-6) ~Q. z(q+ _ q-) IIzll ~
"6'L(6) Ilzll,~ [ I Q(z+q+ - z - q - ) - 6 IQ (Iq+l + Iq-I)].
(2.8)
Observing that
sq(s) >_ MIq(s) l by definition, it follows from (2.6) and (2.8) that
( M - 6)tJlL,(6) f (Iq+l + Iq-I) -< IIh(v)ll~ fJQ ( I q + l + Iq-I), Ilzll ~ .I Q and so
M-6
-((z-((2 .~(6) <_ IIZIv)I1~
for all M < IIz-+Iloo. Letting M I[z-+[1o~/2 and 6 ILz-+11oo/4, we obtain (2.5) and the l e m m a is proved. • The following l e m m a belongs to C h a n g and Liu [4] (see also [5]). =
=
LEMMA 2.3. Let h c C([0, rr] × R 2, R) satisfy (hi), (h2) and (h3)' there are ~ _> 0 and b5 > 0 such that
h(x,t,T)dT>bsIsL c~, I: L~(Q)
H(x,t,s) Suppose that un - u (weakly) in
V Lsl >-~.
and
~Ql4Ix,t,u.)- IQmx, t,.).
(2.9/
I H(x, t, Un - u) ~ O. Q
(2.10)
Then
Remark2.1. I f ~ = 0 in (h3)' then (2.10) implies un ~ u in L ".
270
D I N G Y A N H E N G and LI S H U J I E
In the following, let N~ = L c' 0 N~,, i.e. L ~ = N~ • N~. For u ~ L ~, write u = v + z with v E N ~ and z E N~. LEMMA 2.4. Let h 6 C ( Q x R 2, R) satisfy (hi) - (h2) and (h3)' with ? = 0. Then (i) for each v E N~, inf ~QH(x,t, v+ z) z E N~
is attained at a unique z = z(v); (ii) z ( v ~ ) ~ z ( V ) in L c' if v~ ~ ~; in L ~.
Proof. (i) is clear since the functional z ~ ~QH(X, t. v + z) is strictly convex by (h2), and coercive by (h3)' (P = 0). We verify (ii). Assume v, ~ ~ in L ~'. Since
f H(x,t, vn)>_ f H(.,,t, v~ + z(v,,))_> bsllv, + z(v.)ll~, J
-PQ
Q
by (hi), (z(v~)) is b o u n d e d in L '~. Thus z ( v ~ ) - r in L " a l o n g a subseguence. Since
~QH(X, t,
~'n
+ zG')) >- f H(x, t, )
Q
1,'n
+ z(v,,)),
We have
[oH(X,t, V+-(-~)) > lim f o H ( x , t , vn + >_ lira f H(x, t, v,, + z(v~)) t ? ~o o
JQ
>_ [ H ( x , t , ~ + 2 ) Q by the weakly lower semlcontinuity, and so 3 = z(~) by the minimality, and
lim IQH(x,t, vn + z(v,,)) = f Q H ( x , t , ~ + z(~)).
n~oo
N o w by L e m m a 2.3, noting that v~ + z(v,,) - T + z(T).
I H(x, t. (v,
+
z(v,,)) - (v+
2(i7)))
0.
O and then by Remark 2.1,
H(v,, - ~) + (z(v,) - z(i:))ll~ - 0 which implies (since vn - ;;) z(vn) - z(~), proving (ii). 3. T H E S T R I C T L Y I N C R E A S I N G C A S E
In this section we prove Theorem 1.1. T h r o u g h o u t the section we assume that f satisfies (fl). Let x be the Banach space given by (2.1) with the n o r m (2.3) and the direct sum decomposition (2.2). Set X ~ =Xff, X 2 = X ~ X ~ ~N~ (3.1)
Periodic solutions of a superlinear wave equation
271
if case (1) or case (2) o f Theorem 1.1 occurs, and
);fl = x f ~ X 0,
X2 = X b ~ Ne~
(3.2)
if case (3) o f Theorem 1.1 occurs. Then
X = X l ~ X 2. Suppose that x, l c
c
. . . c x ',
x?
c
c
. . . c x e
are two sequences o f finite dimensional subspaces such that
X j = [..JX~, j = l , 2 . nEN
In what follows, two multiindices 0 = (0 I, 0 e) and r = (V 1, -r e) E N z are denoted by 0 _< r if 01 _< r l, 02 _< r e, and a sequence (0,) c N e is said to be admissible if for each 0 E N ~ there exists m ~ N such that 0n >-- 0 whenever n >__m. For 0 E N 2, let xo
= xJ,
xge.
For a functional • 6 C1(X, R) we denote by ~0 = ~lxo the restriction on Xo. Recall that • is said to satisfy the condition (PS)* if every sequence (uo.) with (0.) being admissible such that t
uo. E Xo., s u p S ( u 0 . ) < oo and cbo.(uo .) - 0 as n - 0o n
contains a subsequence converging to a critical point o f ~. In addition, • is said to satisfy the local linking condition at 0 if for some p > 0 • (u) >_ 0 whenever u E X 1 with [lull - p, • (u) _< 0 whenever u 6 X 2 with Hull < p. The original local linking condition and a r g u m e n t was introduced in [6] under a stronger assumption. Silva in [7,8], Brezis a n d Nirenberg in [9] have used this improved version. Clearly, in this case, 0 is a trivial critical point o f ~. Recall that a function u E X n L °° is a weak solution o f (W) if for all (io ~ X,
~Q[uXCPx - utcPt - f ( x , t, u)qg] = 0. Formally the integrand for
I x2 - u~2 _ F ( x , t , u ) ) ] • (u) =- ~Q [~(u is the Lagrangian for our problem. However it seems to be difficulty for us to treat • as a functional on X since there is no upper b o u n d on the growth o f f as lu[ ~ co. Hence we need to modify f . For k E N with k >_ ? (? is the constant in (f3)), set
f~(x,t,~)=
f ( x , t, ~) min{f(x,t,g),l+
f(x,t,k)+Rk(~ a-l-ka-l)} m i n { f ( x , t, g), - 1 + f ( x , t, - k ) + Rk(k "-L - ( _ ~ ) a - l ) }
for 1~51--- k, for ~ > k , for ~ < - k ,
272
DING YANHENG and L1 SHUJIE
Rk > 0 to be determined. Let gk(x,t,~)
= fk(x,t,~)-
b~,Fk(x,t,~)
= ~ o f k ( x , t , s ) ds and
Gk(x, t, ~) = ~o~gk(x, t, s) ds. Clearly Ifkl-< [fl
for all k,
(3.3)
and (fs) Ifk(x, t, ~)1 -< b ll~l + b2l~l =-1 for all x, t, ~, where b2 depends on k. It is also clear that fk satisfies (fl), (f2), (f4). For Rk first we require it to satisfy Rk >- max
{l+f(x,t,k) k,_l
1-f(x,t,-k)} k,_l
for all (x. t) c Q.
An explict computation shows that fk satisfies (f~) with the same constant ?. We then further require Rk to be chosen so that Ifk-x[-< Ifkl
forall k.
(3.4)
To see this, inductively, suppose we have already found Rk t- By definition,
fk(x, t, ~) = f ( x , t, ~) > fk(x, t, ~) whenever k _< ~ < k + Yk with some Yk > 0 small and independent of Rk. Let now Rk be such that
f ( x , t, k) + Rk(~ ~-t - k c'-I ) > f ( x , t, k - 1) + Rk-l ( ~ - l
_ (k - 1) a-1 )
for all ~ >_ k + Yk. With this choice of Rk one can easily verify that fk >- fk-1 for ~ _> 0. A similar argument for ~ < 0 gives (3.4). Finally, by (f~)-(f4), it is not difficult to see that there are positive constants b3 and b4 both are independent of k, (since ? does not depend on k and we only consider b > 0) such that
fk(x, t, ~)~ > aFk(X, t, ~) - b3,
V(x, t, ~)
(3.5)
and
Fk(x,t,~) >--b41~l ~,
V(x,t,~).
(3.6)
We consider the functional dPk(U ) = ~l fQ(Ig2x -- 122) - f O Fk(x,t,U) l
-
2-bu2) = 2 QI 1 = ~Q(u) - ~g(u)
J Q Gk(x't'u)
where Q(u) = 5Q(U2 - u~ - bu2). It is well known, under (f~)-(fs), that ~k ~ C ~(X, R) and each critical point uk of % is a (weak) solution for
Du = fk(x, t, u).
(3.7)
Moreover, Uk is also a solution of (W) if [[uklloo < k. Therefore there are two things to be done: first, to look for nontrivial critical points of ~k, and then to establish some bounds, independent of k for k large, on the critical points. LEMMA 3.1. ~k satisfies the (PS)*.
Proof. For notational convenience we denote (P = qbk in what follows. We will also denote by dg different positive constants.
Periodic solutions of a superlinear wave equation
273
Suppose uo. ~ Xo., sup, ~(u0.) _< dl and ¢b'.(uo.) = ¢b'o.(uo .) ~ O. Then by (3.5)-(3.6), for n large, writing u = uo., dl + Llull --- ~ ( u ) - ~t (~'n(U), U) _> d2 f o F k ( X , t, U) - d3 > d4llullg
-
d3.
and so ds(1 + Ilullt/% Since dim X° < oo, all norms on X° are equivalent, we see by (3.8) ilullo,
(3.8)
-<
Ilu°ll _< d6(1 + Ilull~/%.
(3.9)
Since Na is closed in L a with a closed complement L~ -- clL. (X~- • X ° • X ~ ) (where ClL. (S) denotes the closure of the set S in L~), the projector along L~' onto No, is bounded, and it follows from (3.8) that Ilzll. -< d7(l + Ilu111/%. (3.10) Noting that Q(u) is positive definite on X~ and I1( Q ' ( U) , U+ ) = (¢b'n(u), U+ ) + ] 0 g k ( x , t , u ) u +
we get IIu + IIz _< ds IIu + II + d9 Jo Igk(x, t, U) II U+I which, jointly with (f5), yields (o( = c¢/(c¢- 1)) Ilu÷ll2 -< dsllu+ II + dl0
(fo
(lul + lul ~-~)
Ilu+ll~
-< all(1 + Ilull~-l)llu+ll. Combining this and (3.8) implies Ilu+ll ~ d12(l + ltull<~-l/:%.
(3.11)
Ilu-lL ~ d13(1 + Ilull<~-l>:~).
(3.12)
Similarly we also have It then follows from (3.9)-(3.12) that Ilull ~ d14(1 + ilull~-l)/%. Hence (uo.) is bounded. Recall that X~ are compactly embedded in L ~ and dim X° < co. By the boundedness of (uo,), one can assume that u ° ~ -° u in X, vo, - Uo-" + u ° + u~, ~ -~ in X , vo, - -~ in L ~, and zo, - -g in L ~. Since Xb+- and X° are Hilbert spaces and N~ has a Schauder basis, we may assume that the subspace sequence (Xo.) has been chosen so that the associated projection operator sequence (Po.) is bounded, i.e. II1'o. II <- const., and (I - Po,)u - 0 for each u. Since ~',(uo,) - O, zo, = Po, zo, and, according to Lemma 2.4 (recall that f k satisfies all the assumptions of the lemma), z(vo.) ~ z(V), f Q f k ( x , t, I:O, + ZOn) (Zo. - Z(~on) ) = ( ~ ' ( u o . ) . ZO. - z(vo.)) = (¢b'n(uo.),zo. - z ( v o . ) )
- (~b'(uo.), ( I - P o . ) z ( v o . ) )
~ O.
274
D I N G Y A N H E N G a n d LI S H U J I E
Hence passing to the limit in the inequality
IQFk(x't'v°"+z(v°'))-~QFk(X't'v°"+z°°)>f
fk(x't'v°'+z°")(z(v°')-z°")Q
one o b t a i n s
I
Q
Fk(x,t,f+z(f))
> lim
n-e~
Q
& ( x , t , vo,+zo,,)
-> ~olimf Q & ( x , t, vo, + zo,)
>__~QFk(x,t,-~ + 2) > j'QFk(X, t,-~+ z(7)) a n d so ~ = z(~) a n d
IQFk(x,t, vo, + zo,) -- ~QFk(x,t, v + z(v)). N o w a p p l y i n g L e m m a 2.3 yields I1% +zo, - 5 + z(5)[[~ - 0 a n d zo° ~ ~ = z(V). Therefore uo° ~ V + z(V) in L ~, a n d a s t a n d a r d a r g u m e n t shows that W(uo,) - W ( ~ ) = ' g ' ( 7 + z(V)). Since O',(uo,) ~ O, it follows that Q'(uo,)/2 ~ W(K) which implies u~, ~ ~± a n d so uo, ~ ~ in X . Finally, it is easy to see that ~ is a critical p o i n t o f qs, a n d the conclusion follows. LEMMA 3.2. qSk satisfies the local linking c o n d i t i o n at 0.
Proof. First a s s u m e case (1) o r (2) o f T h e o r e m 1.1 occurs. M o r e o v e r we consider only case (2) since case (1) can be treated similarly. Recall that in this case X 1 = Xb+ given by (3.1). N o t i n g t h a t Q(u) is positive definite on X 1, we have 1Q(u) > dlHul] 2,
V u E X 1.
(3.13)
By (f4) Gk(x,t,~) = o(l~L 2) as ~ ~ 0 u n i f o r m l y in (x,t) ~ Q, a n d so for any E > 0 there is d~ > 0 such t h a t [Gk(x, t, ~)[ < E~ 2 + d~l~[ ~ which implies t h a t [~(u)[ < Ed21[ul[2 + d31lul[ ~, or 't'(u) = o ( l l u f )
as
u ~ 0
(3.14)
It then follows f r o m (3.13)-(3.14) t h a t there is a p > 0 so t h a t I rb(u) = rbk(u) >_ ~dlllul[ 2 _> 0 V u E X l C~Bp
(3.15)
rb(u) <_ O, V u ~ X 2 = X ° • Xd- • Nc, with Ilull -< p
(3.16)
N o w we prove p r o v i d e d p being small e n o u g h . T h e a r g u m e n t a n a l o g o u s to t h a t o f [2, P r o p o s i t i o n 3.1]. F o r the reader's convenience, we sketch the proof. Let u = u ° + u - + z ~ XZ • Xb- • N~ a n d C = s u p { l , ( u ) "u ~ X 2 n B o } .
Periodic solutions of a superlinear wave equation
275
Since ~ l x ~- is weakly u p p e r semicontinuous. So the s u p r e m u m is attained at some ~ E Bp. If Ilu-II = p and p is small enough then one immediately get
cI,(u)<_O-~ep t
2
<0
VuE
X
2
n B- p
for some E > 0. Hence Ilff-II < P. This implies that t~(P^u") = . P ^ A ( x , t,
in the sense o f distributions where A = {(j, k) E N x Z ; j 2 - k 2 < 0}. F r o m L e m m a s 2.1 and 2.2 we get that I1~11oo -< I1~°11 + lib'- Iloo + IlYtloo -< 6 i f p is small enough, where 6 was given in T h e o r e m 1.1. Therefore from that G(x, t, ~) >_ 0 as I~1 -< 6 we get ¢b(~ <_ - ~ Q G ( t , x , u ~ d x d t
<_ O
this proves (3.16). In the case (1) we do not require that G(x, t, ~) >_ 0 as I~1 -< 6 because @(u") ___ -dallH- 112/2 - bll~ll~/2 + e(p)llHIl~ -< 0 as p small• In case (3) X 1 = Xb+ • Xb~, X 2 = Xb- • N , and let C = inf{~(u) • u E X l n Bo}. An argument analogous to that showing (3•16) gives ~ ( u ) >_ 0 V u E X 1 n Bp. For u E X 2 consider sup{~(u) • u E X 2 n Bp} and the p r o o f is similar to case (2). P r o o f o f Theorem 1.1. If for each k there is a critical poiut uk u: 0 0 f @ k such that ~k(Uk) < c where c is independent o f k then it is easy to prove that Ilukll~ is b o u n d e d and therefore Uk ~ 0 is a solution of (W) if k is large enough. (a) Suppose that there is a k0 such that Yp - @k0 has no critical point u ~ 0 with @(u) _< 0. By the argument to that [10, T h e o r e m 2] we can define a m a p p i n g y • aQ~ - x,,~1 +1 ~ )(2 such that ~(y(s, u)) _< 0 for all (s, u) E ~Q, where Q, = [0, 1] x (X 2 n B p ) and ml > 0 is a positive integer. F r o m Ifk01 < Ifkl as k >_ k0 it follows that ~k(y(s, u)) _< 0 for all (s, u) E OQ, as k >_ k0. Let F = {~, e C ( Q , , xtj) " ~I~Q, = z} where/3 = (n, n) with n _> ml + 1. Choose Pk SO that ~k > 0 on X l n Bp,. It has been shown in [10, T h e o r e m 2] that for each sufficiently large n one m a y deform X2 n Spa to a set Sk C X2 such that @k >-- Ck > 0 on Sk where Ck is independent o f n and the sets ¥(aQn) and Sk link nontrivially. Let G ( k ) = inf
sup
~k()'(s,u))
~EF (s,u)EQ,,
Since "~ 1 I C -= sup{@(u) " u E X,~,+ 1 • X 2 } >- sup{dPk(U) " u E A"~, +1 • X 2 }
therefore Ck <- Cn (k) <_ C where C is independent o f k. A s t a n d a r d a r g u m e n t show that C,(k) is a critical value of (~k)a where/3 = (n, n). Using (PS)* condition and letting n - oo we get a critical poiut Uk of CI'k with 0 < Ck <- ~k(Uk) <-- C. It implies that Uk ~ O, Ilukll~ is b o u n d e d and Uk is a solution o f • as k is large enough. (b) Suppose now that k0 stated in (a) does not exist. Then for each k there is a critical point Uk ~ 0 of @k with @k(Uk) --< 0. This implies Uk ~ 0 is a solution o f (W) if k is large enough. T h e p r o o f is complete. •
276
D I N G Y A N H E N G and LI SHUJIE 4. THE N O N D E C R E A S I N G CASE
F r o m now on we always suppose that all the assumptions of T h e o r e m 1.2 are satisfied. Recall that we are only interested in the case b > O. LEMMA 4.1. For each k >_ ?, there are Pl > 0, A > 0 depending on k, and P2 > 0 independent of k, such that (~k(U)>_O V uEXINBp~, ~k(U)>~ V uEXIAOBp~, (4.1)
• k(U) --< 0 V u C X 2 n Bp2-
(4.2)
Proof. Recall that if case (2) of T h e o r e m 1.2 occurs, then X l = X~+ and X 2 = X~ • Xff • Nc~. (4.1) is clear by the first part of the p r o o f in L e m m a 3.2. We consider (4.2). Since g(x, t, ~)E > 0 whenever IEI <- 6, Ifk(x,t,E)] > ]bEI iflE[ -< 6. Recall that for IE] >- E fk(x, t, ~)E >- Fk(X, t, ~) >_ dl~l ~, and so fk(x, t, ~) >_ d~ ~-1 for ]El > ?, and fk(x, t, E) <- - d ( - ~ ) ~-1 for ~ _< -?. Let .f(~) ~ C(R, R) be a strictly monotonically increasing function given by aT(~ ) =
and f ( ~ ) = - f ( - ~ )
[
b~ ll(E) 12(~) dEC,-1
if 0 _< E -< 6/2, if 6 / 2 < E <~, ifff 27,
for ~ < 0, where
if-
b6 +~. 2d(2~)c,-2
ll(E) is a linear function determined by /l(~i/2) = b(6/2), 11(7~) = b6, and /2(E) is a linear function determined b y / 2 ( x ) = b6,/2(27) = d(27) ~ i Then Ifk(x, t, E)I > [Y(~)I, and
& ( x , t, ~) >_ IF(E) = I ~ f ( s ) ds,
for all
~ - ~O F ( ~ ) . Then Let g)(u) = ~l ~o(U;.~ - UT) cI,~(u) _< gP(u)
for all u E X.
By the second part of the p r o o f for case (2) in L e m m a 3,2, there is a P2 > 0 such that g~(u)_<0,
V u~X
2nBp:
and (4.2) is proved in this case. If case (3) of T h e o r e m 1.2 occurs, then X I = Xh+ • X~, and X 2 = X~- • Na. Set fk(x, t, ~) = fk(x, t, ~) + A2o¢I~I~-2E. Then
-~k(x, t, E) = Fk(X, t, ~) + A2{~I ~ --= ~E 2 + (Gk(x, t, E) + A21EI% > Fk(X, t, E),
'I e (u2-u~)-I o i~(x' t, u)
g)k(U) = ~
= ~(u)
- A211ullg -< ~ ( u ) .
Periodic solutions of a superlinear wave equation
277
Since ~ ( x , t, ~) is strictly increasing in ~, and (Gk(x, t, ~) + A21~I") -< 0, by the first part of the p r o o f for case (3) in Lemma 3.2, there is a Pl > 0 such that • k(U)>-YOk(U)>O
V ueX
1c~,.
We show that ¢bk[Xt~S~ >_ A for a A > 0. If not, there are u~ ~ X ~, ][u~ll = p~ such that
0 <_ ~k(U.) = ~k(u.) + A2llu, ll~ ~ 0. Then I[u~ [I ~ 0 and so HUn + I[ - P~. We see that ]
+
[
'l'k(U.) = ~Q(u. ) - j >- ~ Q(u.+ )
-
+
Q
Gk(X, t, u. + u~)
Cl Ilu.[l~
c211u.ll~.
the left-hand side tends to 0. while the right-hand side tends to a positive constant, a contradiction, (4.1) follows. Next, to prove (4.2), we again define a function f ( ~ ) ~ C(R, R) by
=
and f ( ~ ) = - f ( - ~ )
b ~ - h l p ~ °-I
for 0 _ ~ _< 6/2
Ii (~) /2(~) d~ ~-1
for 6 / 2 < ~ <_ for 2 < ~ < 2? for ~j ___2?
for ~ < 0, where
6(b - pal (~)p-2) X=
+?,
4d(2?)a-:
II (~) is a linear function determined by Ii (6/2) = (b - pat (6/2) p-2) ~, Ii (-£) = ( b - pAI6p-2)6, and l:(~) is a linear function determined by 12(~) = II (~), 19(27) = d(27) " - l . Then it is easy to verify that f ( ~ ) is strictly increasing, fk(x, t, ~) = b~ + gk(x, t, ~) >--f ( ~ ) , and
Fk(X, t, ~) >__ff(~) =ff(~) =
~2 + ( - h l [ ~ f )
f ( s ) ds, if I~1 <-
~.
By the second part of the p r o o f for case (3) in L e m m a 3.2, one sees that there is a P2 > 0 such that
• k(u) <-~(u) - 1 fQ(u2_u2t) - f Q f f ( u ) < O , The lemma is proved.
V uEX2n-Bm.
•
N o t e that since Ifk-I (x, t, ~)1 --< Ifk(x, t, ~)1, without loss of generality, we can assume that A(k) _< ~k(u) <- A,
V u E X t n OBp,
(4.3)
In this section the notations X, X~, X~,, X b, N~, Xn, Xo, Po still have same meaning as before. Since 79 - { s i n j x s i n k t , s i n j x c o s k t ; (j, k) E N × Z} is a Schauder hasis for X,
IIPo.[l£Ix,x) <- Mi
(4.4)
278
DING YANHENG and LI SHUJIE
for some M1 > 0, and ]l(l-P0,,)u[]
-0
as
n-
~
(4.5)
for each u ~ X. Noting that, for any q ~ (1, ¢~), :P is also a basis sequence in Lu which can be extended to a Schauder basis for LU Po, can be extended as a projector from L u onto Xo,, such that IIPe~IIz 0 depending on q. L e t N T = {k ~ N; k >_ F}. LEMMA 4.2. If, for an admissible multiindex sequence (0,) c N 2 , there is an infinite subset A c NT such that, for each k ~ A, there exist p = p ( k ) > 0 and uo,, ~ Xo,, satisfying Ilu0,1l >- p, cbk(ue,) _< A + 1, (~k)o, (uo,,) = 0. then (W) has a nontrivial weak solution.
P r o o f For notational convenience, write ff = ~k, 4% = (cbk)o,,, and let d, denote positive constants, which are independent of k, and c(k) denote positive constants depending only on k and varying with the places. Since ~b(uo,) _< A + 1 and cb'~(uo,) = 0. (4.7) we have
A + 1 > ¢b,(uo,) - 2i (~'.(uo,,), uo°) f = Jo{½fk(x, t, uO,)uo, -- Fk(x, t, UO°)] which, together with (3.5)-(3.6), implies
IO Recall that, f o r u E X ,
Fk(x't'u°")<-~Q fk(x't'u°")u°°<-d'
(4.8)
tlu0,ll~ -< d2.
(4.9)
u = u + + u ° + u- + z - v + z. By (4.7),
Q(u~ ) = (~I Q, (uo,),uo,> + = fQ g k ( x , t , uo,)u~. +
(4.10)
Since Q(u) Is positive definite in X# and negatine deginite in X # , (4.10) and (J~) as well as (4.9) yield Ilu~,,ll -< c(k)llu~o[l~,. (4.11) Noting that dim Xh° < oo by (4.9) Ilu~,ll -< d31lu~,l], -< da[luo°ll,~ <- ds.
(4.12)
Since Na is a closed subspace of L " with the closed c o m p l e m e n t clL~ ( X ) , the closure of X in L ~, the projector from L " onto Na is bounded, and so IIz0, ll -- Ilzooll~ -< d611uooll~, <- d7 by (4.9). N o w (4.1 1)-(4.13) show that Iluo, II ~ c(k).
(4.13)
Periodic solutions of a superlinear wave equation
279
Therefore, without loss of generality, one can assume
uo,-~k
in X,
vo, ~-fk in L ~.
(4.14)
In virtue of (4.7)
Duo. = Po, fk(x, t, uo,)
(4.15)
and consequently, it follows from (4.6), (fs) and (4.9) that IIDu0.11., = NPo.fk(x, t, uo.)ll., _< M . , Ilfk(x, t, uo.)ll.' -< c(k). where a ' = a / ( a - 1). Hence one can assume Duo. - ~ in L ~'. Since, by (4.14), Duo. ~ D-~k in the sense of distributions, g = t3~k. N o t e that for each qo E X,
I
Q
+ fQ(fk(x, t, uo,)(I - Po,)cP
(4. 16)
>-- j (fk(x,t, uo,)(I-Po,)q). Q Since I1(I - Po.)q)ll - 0 (see (4.5)) and by (4.14)
IQ °u°" " u°" = IQ rTu°" " v°" -- ~Q D ~ " ~k = ~ r T ~ k ' ~ passing to the limit in (4.16) yields
O(O~k - f k ( x , t, qo))(Kk - qg) >__0.
(4.17)
Inserting in (4.17) the qo = uk + sx, where s > 0 and X E X, dividing by s and letting s ~ 0 give Q(D-iik -- f k ( x ,
t, -ffk))X >-- O.
(4.18)
Since X was chosen arbitrarily, (4.18) shows that ~k is a weak solution for
D-ak = fk(x, t, ~k).
(4. 19)
N o w we show that Kk ~: 0. Arguing indirectly, assume Kk = 0. Then yD, ~ 0 in L a, which, jointly with (4.11), implies vo, ~ 0 in X, and
dlluo, ll~, <- ~QFk(x,t, uo,) <- IQfk(x,t, uo,)Uo, = fQmuo, • YD, <<-c(k)llvo, ll~ -- O. Therefore IIz0.11a ~ 0, and then we get 0 < p _< Iluo.II -< Ilvo.II + Ilzo. II - 0.
a contradiction.
280
D I N G Y A N H E N G and L1 S H U J I E
Next we establish bounds for Iluktlo~. Note that, by (4.19)-(4.15),
Jef*(x't'~*)~*=Je
(m~,-=uo,,)v,+fe
=,oo(V,-vo,)+fef,(x,t,,o,,),o
(4.20)
Since the first and second terms of the right-hand side of (4.20) tend to 0 (as n ~ ~ ) , by (4.8).
IQf,
(x. t.~,)~- <_ d~,
and so
>1
fk(x, t, K~)-Kk + I
]~_<1
IJ~(x, t, ~ ) 1 _< ds.
Hence, by (4.19) according to [11] or [1], II~k[Io~ ~ d9llfk(x,t,~k)lll ~ dlo. Since fk satisfies (f~), for each R > 0, there is dR > 0 depending on R but independent of k such that ]fk(x, t, g + ~) - fk(x, t, ~)l -> dRlg[ - 1 whenever Igl -< R. Taking into account [2, R e m a r k 3.2] with R = dl0, we get
Ilrkll~ -< dll where ~k = ~k + ~ . Hence I1~11~ -< d12. Therefore for k large, say, k >_ d ~ + 1, ~ is a nontrivial weak solution of (W). The lemma is thereby proved• •
Remark 4.1. In virtue of L e m m a 4.2, to prove T h e o r e m 1.2, it remains to consider the case that there is a k0 >- ~ such that, for every p > 0 and each k >_ k0, there exists 0% = ocp(k) = (mp, mp) ~ N 2 satisfying, for any o¢ >_ ocp (~k)'~(u) ~e0,
V uEX~
with ]lull >-p and ~ k ( u ) - < A +
1.
Proof of Theorem 1.2. Let ~ = ~k0 (see R e m a r k 4.1). Let P2 be the constant in L e m m a 4.1. By R e m a r k 4.1, there is mp2 = mo2(k) such that for ¢x >_ O~p2 = (mp2, mp:), ~'c~(U) "~ O, Vu c Xc, with Ilul[ -> P2 and ~ ( u ) _< A + 1. It follows from (f~) that i f R is large enough, then ~ ( u ) _< 0 for each u ~ X m,2+l ~ @ X 2 with I[u[[ >- R/~/~. Let e~ = (mp~_,n), n >_ mp:+l. Since dim Xc,- R. Denote
Q. = [0, 1 ] x ( x 2 n ~p: ) and define a m a p p i n g y • 3Q~ - X , oo~+~X~by 1 f o r 5" = 0
y(s, u) =
o-(2s, u) 2(1 - s)cr(1, u) + ( 2 s - l)v0 V0
b/ C 2 2 f-'l Bp2,
fors~(0,½] 1
for s e (~, 1) for s = 1
u~X }n&2, u E X} n Sp2, ucX 2nBp2.
Periodic solutions of a superlinear wave equation
281
Then ~(¥(s, u)) __ 0 for all (s, u) E OQn. Since Ifk01 --- Ifkl i f k _> k0, it follows that ~k(y(s, u)) __0 for all (s, u) E ~Qn whenever k >_ k0. Let/3 = (n, n) whenever n > mp2 + 1, Yl~e.= Y}
F = {~ E C ( Q . , X o ;
and cn(k) = inf
sup r~k(~'(s, u)),
Vk >_/co.
Let p~ = Pl (k) be the constant in Lemma 4.1 so that • k(u)->0 on X l n B p , ,
~ k ( u ) > - - A = A ( k ) on X lnSp~.
By Remark 4.1, for all a >_ ap~(k), ('bk)'~(u) * 0 for u ~ X~, Ilull >- Pl and '~k(U) <-- A + 1, particularly, for all u E X1 n So, with a = (n, n) >__apl (k). Therefore we may deform X.l n So, to a set S c X., using a pseudo-gradient flow for (~k)~, such that ~k(U) >- A ( k )
V u E S,
(see [10, Lemma 1]). Moreover, along a line in the proof of [10, Theorem l, step (3)], one can show that y(aQ,) and S link (i.e. ~(Qn) n s * qb for any ~ E F). Therefore C.(k) > A(k). 1 1 In addition, as y(0Q.) c X,.~2+l • X2, there exists a y0 c F with y0(Q.) c X,~.2+l • X2. Since 1 d = s u p { ~"~( u ) ; u E X.rap2 l + I ~ X2} > sup{d~k(U);U E Xmp2+l O X'2},
we have C.(k) <_ d
(d independent of k). A standard argument shows that C.(k) is a critical value of (CI'k)t~(/3 = (n, n)). Let uo(k) ~ XO be the associated critical point of (~k)~. Then A(k) __<~k(U~) --< d,
(~k)'(u~) = 0.
Repeating the procedure for proving Lemma 4.2, one sees there is a weak solution fik for t3~k = f k ( x , t, -Ok).
and II~kll~o --< dl independent of k. Now we show that ~k :~ 0. If not, one can verify that Ilu~ll- 0 in X. Then we obtain that A(k) _< '~k(U~) -- 0 as /3 ~ 0% a contradiction. Finally, since II~kllo~ -< d~ independent of k, for k large, Uk is a nontrivial weak solution of (W) The proof is thereby complete. •
282
D I N G Y A N H E N G and LI S H U J I E REFERENCES
I. R A B 1 N O W I T / . t: 1t., Free vibrations tor a semilinear wave equation. Comm. Pure Appl. Math., 1978, 31, 31 68. 2. LI. S. J. & SZI ,41N A . Periodic solution for a class of n o n a u t o n o m o u s wave equations differential and integral equations, to be i ' ~,i bed. 3. W I L L E M , I~!. ,, ~ h ~monic oscillations of a semilinear wave equation. Nonlinear Analysis, 1985, 9, 503 514. 4. C H A N G , K. * .~ ' ~J. J. Q., The existence of infinitely m a n y solutions lbr the semilinear wave equation, to be published. 5. D I N G . Y. ,L, A l, ' on nonlinear beam equations (in Chinese) Acta Math Sinica. 1990, 33, 172 181. 6. LIU. J. Q. & I I Some existence theorems on multiple critical points and their applications. Kexue Tongbao, 1984. 17. 7. SILVA. E. ~,. B ~r:I'cal point theorems and applications. P h D. thesis, Univ. of W i s c o n s i n - - M a d i s o n , 1988. 8 SILVA. E. A. B r - :ng theorems and applications to semilinear elliptic problems at resonance. Nonlinear Analysis, 1991. 16. 455 -* ' 9. B R E Z I S . It. & " ' . N B E R G , L.. Remarks on finding critical points. Comm. Pure Appl. Math., 1991.64, 939-963. 10. LI. S. I & '~\ l[.J .,i. M.. Applications o f local linking to critical point theory. J Math. Anal. Applie., 1995, 189, 6-32. 11. BREZI'~. it.. COtv~ ': J. M. & N I R E N B E R G , L.. Free vibrations tor a nonlinear wave equation and a theorem of P. Rd~'!;~owJt7 e ,.,n Pure dppl Math., 1980.33, 667 689
Pergamon
PII: S0362-546X(96)00045-4
PERIODIC
SOLUTIONS
OF A SUPERLINEAR
WAVE EQUATION*
D I N G Y A N H E N G and LI SHUJIE Institute of Mathematics, Academia Sinica, 100080 Beijing, ER.China
(Received 19 December 1995; receivedfor publication 28 March 1996)
Key words and phrases: Periodic solution, wave equation, minimax method.
1. INTRODUCTION
We consider the existence problem of periodic solutions for the following wave equation { utt - Uxx = bu + g(x, t, u), x E (0, rr), t E R, u(O, t) = 0 = u(rr, t), t E R, u(x, t + 2rr) = u(x, t), x ~ (0, rr), t ~ R.
(W)
Assume that f (x, t, ~) - b~ + g(x, t, ~) satisfies (fl) f E C([0, rr] × R 2, R) is 2rr-periodic in t; (fE) f ( x , t, ~) is strictly monotonically increasing in ~; (f3) there exist ? > 0 and ct 6 (2, co) such that O
f(x,t,s) ds<_f(x,t,~)~,V(x,t,~)witht~l>-_-i,
(f4) g(x, t, ~) = 0(1~1) as ~ - 0 uniformly in (x, t). Let rn = a2/Ot 2 - aElax 2, and o-(n) the set of eigenvalues of [] subject to the periodic and boundary conditions in (W). Set G(x, t, ~) = ~ g ( x , t, s)ds. We shall establish the following results: THEOREM 1.1. Assume that f satisfies (ft)-(f4). Then (W) possesses at least one nontrivial (weak) solution provided moreover that one of the following cases holds: (1) b ~ tr([]) or b = 0, (2) b ~ o-([]) and there is a 6 > 0 such that G(x, t, ~) >-_0 whenever I~l - 6, (3) b ~ o-([]) and there is a 6 > 0 such that G(x, t, ~) <_ 0 whenever [~l -< 6. THEOREM 1.2. Assume f ( x , t, ~) satisfies (fl), (f3), (f4) and (f~) f ( x , t, ~) is nondecreasing in ~. Then (W) possesses at least one nontrivial weak solution provided moreover that one of the following cases occurs: *This research was partially supported by the Chinese National Science Foundation. 265
266
D I N G Y A N I l E N G and LI S H U J I E
(1) b ¢ o-([3) or b = 0; (2) b E if([]) and there is a 6 > 0 such that g(x, t, ~)~ >_ 0 whenever I~l ~ 6; (3) b ~ o-(E3) and there are 6 > 0, 2~1 > 0, he > 0, 2 < p _< o~ such that - h i [~[a _< g(x, t, ~)~ <_ - A21 ~ I", whenever t ~1 -< 6. Note that f2 (or (f~)) and (f4) imply b _> 0. T h e o r e m 1.1 and 1.2 remain valid under the same hypotheses if (W) is replaced by
u, - u~-,. = - ( b u + g(x, t, u) ) T h e o r e m 1.1 and 1.2 generalize Rabinowitz' result [1] and a result of [2]. In [1] Rabinowitz proved that (W) has a nontrivial solution if b >__0 and ~ - g(t, x, ~) is increasing. In [2] Li and Szulkin had proved T h e o r e m 1.1 but under the strongly monotonically increasing condition for f : there exists an t > 0 such that (f(x, t, ~) - f ( x , t, 0 ) ) ( ~ - 0) >- EI~ - r/I ~ for all x, t, ~, rl. When b = 0, T h e o r e m 1.1 has been proved in [1] and T h e o r e m 1.2 has been proved in [2]. In the present paper we only need to deal with the b > 0 case and we do not ask that ~ ~ g(t, x, ~) is increasing. T h e o r e m 1.2 is m o r e delicate because the variational form of (W) does not satisfy (PS)* condition if f satisfies (f2). O u r p r o o f relies on the approximate m e t h o d and a special m i n i m a x m e t h o d under the local linking condition.
2. P R E L I M I N A R I E S
Let Q = (0, rr) x (0, 2rr) and [I ' lip denote the L P ( Q ) - n o r m for p ~ [1, oo]. We consider the Banach space
X=[u=
(2.1)
j~tq~"ukjsinjxeik';u-~J=-ff~j and llullx < °° t k~Z
with the n o r m
Ilullx
Ij 2 - k 2 - bllukjl 2 +
=
j:-
O,b
~. j:-kZ=b
where cx E (2, oo) is the constant of (f3) and z =
~ . Ukj s i n j x e ikt. j2 =k 2
Clearly X is reflexive. Set
Im:jl 2 + Ilzll~
}'
Periodic solutions of a superlinear wave equation
X~ = [ u c X ;
u=
267
Z Ukj sin j x e ikt , jZ-k2>b ~'. Ukj sin j x e ikt , j2-k2
X~ = f u E X ;
u = j2~k2
x° = {u e x; u=
~"
ukj sin j x e ikt ] ,
j2-k2 =b
{u xu: Ukjsinjxek} /'2 = k 2
where it is understood that if b = 0, X° = X ° = {0}. Then we have the direct sum decomposition
X = X~- ~ X ° ~ X f
~N,
(2.2)
and, correspondingly, u = u + + u ° + u- + z for u 6 X. It is known that X~ are compactly embedded in/_Y for p 6 [ 1, oo), and dim X° < ~. In what follows, for convenience, we introduce the following equivalent norm on X: Ilull -- max{llu+llx, Ilu-IIx, Ilu°llx, Ilzll,~}.
(2.3)
For any index set A C N × Z, let F
Z Ukj s i n j x eikt } (j,k)EA
X^=tu~X;u= and define the projector P^ " X - X/, by
P^u =
~"
Ukj sin j x e ikt.
(j,k)EA
In particular, set/x0 = {(j, k) E N x Z; j2 = / f l }. Note that X^ 0 = Na. LEMMA 2.1. Let h(x, t, s) ~ C([0, rr] × R 2, R) be 2rr-periodic in t and satisfy (ht) there are bl, b2 > 0 such that Ih(x, t,s)l <- bllsl + b2lsl '~-l. Suppose that ^ c N × Z \ ^ 0 and u E X satisfies n(P^u) = P^h(x, t, u) in the sense of distributions. Then IIe^ulloo -< b3llullc, + b4llull~ -1 where b3 and b4 depend only on bl and b2.
Proof. The proof is closely that of [3, Proposition 1] or [2, Lemma 3.1]. We omit the details. Suppose that h ~ C([0, rr] × R 2, R), h(x, t, 0) = 0, and satisfies: (h2) h(x, t, s) is strictly monotonically increasing in s; (h3) h(x, t, s) satisfies (f3). Set, following [1], for L > 0, min ( h ( x , t , s + T ) - h ( x , t , r ) )
I'rl_
e~(s)
=
fors>__0
(x,t)~.-Q max(h(x,t,s+r)-h(x,t,T)) ITI-
(x,t)~Q
fors<0
268
D I N G Y A N H E N G and LI SHUJIE
and /2~"(s) = min (tpjL,(s), - tp~(-s) ). Then tp~(s) and U~(s) are strictly monotonically increasing and tp~'(s) _+~,/2to, (s) - ~ as s - ~ .
-
+_~ as s -
LEMMA 2.2. Let h ~ C([0, rr] × R 2, R), h(x, t, 0) = 0. satisfly (h3) and (h2)' h(x, t, s) is nondecreasing in s. Suppose that u = v + : E X with v E L ~ and : E N~ satisfies 22)(/) -< 0,
f (h(x, t, u) t AIzl' Q
V q) c N,~.
(2.4)
For some A _> 0. Then - ~ L °°. Moreover, if h satisfies (h3) and (h2) instead of (h2)', then for each u = v + z c X with IIvlloo _< L, z ~ N~ and satisfying (2.4), we have 1
t2~(511zllool _< 4tlh(v)ll~
(2.5)
where (h(v))(x, t) = h(x, t, v(x, t) ).
Proof. The p r o o f is similar to [1, L e m m a 3.7] or [2. L e m m a 3.2]. In fact, for any M > 0, define a function q : R - R by q(s) = 0 if Isl -< M, q(s) = s - M i f s > M , and q(s) = s + M if s < - M . Recall that each z E N~ can be represented as z ( x , t ) = p ( x + t) - p ( - x + tl - z + - z with fo" p(-r) d T = 0. where p is periodic with 2rr (see [1]). Set q+ = q(z + ) and q - = q ( z - ) . Then q)=q+-q~ N , , q + >_q- i f z _ > 0 a n d q + <_q ifz<_O, a n d f Q Z q+ = fQz+q - = 0 . By(2.4),
~Q(h(x,t,v+-)-h(x,t,v))(q+-q
-) <_ I Q ( h ( x , t , u ) + A[zl ~ 2 z - h ( x , t , v ) ) ( q + - q <_ - f
JQ
)
h(x,t, v)(q + - q - ) <_ IIh(v)llo~ f (/q+l + Iq-I), JQ
(2.6) where (h(v))(x, t) = h(x, t, v(x, t) ). By the superlinear assumption (h3), for any R > 0 there is a constant CR > 0 such that
]h(x,t,S+T)-h(x,t,T)l Let R = we have
if Ivl _
>-CRIsl-1
vll~o. N o t i n g that, by (h2)', the integrand on the left-hand side o f (2.6) is nonnegative,
~o(h(x't'v+z)-h(x't'v))(q+-q-)>-CR~oZ(q*-q-)-fQ >_ ( C R M -
(Iq+l + iq-I)
1) f (Iq+l + Iq-I), 2Q
which, together with (2.6), implies
(CRM-
1)((Iq+l JQ
+ lq-I) -< IIh(v)ll~ f ( I q + l JO
+ Iq-I).
(2.7)
269
Periodic solutions of a superlinear wave equation If z is not essentially bounded, the integral in (2.7) is positive and so
C R M - 1 _< Ilk(v)Iloo for all M > 0 which is a contradiction. The first conclusion o f the l e m m a follows. Suppose h satisfies (h2) instead of (h2)', and u = v + z with Ilvlloo -< L and z ~ Na satisfies (2.4). I f z = 0, (2.5) is clear. Hence assume z ~ 0. For any 6 _> 0, let Q6 = {(x, t) c Q; Iz(x, t)l >6}, Q~ = {(x, t) ~ Q;z(x, t) >_ ~5}, and Q2 = Q6\Q~. By the first conclusion, Ilzlloo < oo. One has
I
Q;
(h(x't'v+z)-h(x't'v))(q+-q-)
>I
Q;
~JL'(Z)z(q+-q-) >'~IL'(6) I z
and I
Q~
(h(x, t, v + z) - h(x, t, v))(q +
I
q-) >
-Ilzll~
Q;
z(q + - q-).
'I'~(-6)llIe; zlo~
z ( q + - q-)"
Therefore,
I
+ - q-)>_
(h(x,t,v+z)-h(x,t,v))(q O~
>_
~11~(-6) ~Q. z(q+ _ q-) IIzll ~
"6'L(6) Ilzll,~ [ I Q(z+q+ - z - q - ) - 6 IQ (Iq+l + Iq-I)].
(2.8)
Observing that
sq(s) >_ MIq(s) l by definition, it follows from (2.6) and (2.8) that
( M - 6)tJlL,(6) f (Iq+l + Iq-I) -< IIh(v)ll~ fJQ ( I q + l + Iq-I), Ilzll ~ .I Q and so
M-6
-((z-((2 .~(6) <_ IIZIv)I1~
for all M < IIz-+Iloo. Letting M I[z-+[1o~/2 and 6 ILz-+11oo/4, we obtain (2.5) and the l e m m a is proved. • The following l e m m a belongs to C h a n g and Liu [4] (see also [5]). =
=
LEMMA 2.3. Let h c C([0, rr] × R 2, R) satisfy (hi), (h2) and (h3)' there are ~ _> 0 and b5 > 0 such that
h(x,t,T)dT>bsIsL c~, I: L~(Q)
H(x,t,s) Suppose that un - u (weakly) in
V Lsl >-~.
and
~Ql4Ix,t,u.)- IQmx, t,.).
(2.9/
I H(x, t, Un - u) ~ O. Q
(2.10)
Then
Remark2.1. I f ~ = 0 in (h3)' then (2.10) implies un ~ u in L ".
270
D I N G Y A N H E N G and LI S H U J I E
In the following, let N~ = L c' 0 N~,, i.e. L ~ = N~ • N~. For u ~ L ~, write u = v + z with v E N ~ and z E N~. LEMMA 2.4. Let h 6 C ( Q x R 2, R) satisfy (hi) - (h2) and (h3)' with ? = 0. Then (i) for each v E N~, inf ~QH(x,t, v+ z) z E N~
is attained at a unique z = z(v); (ii) z ( v ~ ) ~ z ( V ) in L c' if v~ ~ ~; in L ~.
Proof. (i) is clear since the functional z ~ ~QH(X, t. v + z) is strictly convex by (h2), and coercive by (h3)' (P = 0). We verify (ii). Assume v, ~ ~ in L ~'. Since
f H(x,t, vn)>_ f H(.,,t, v~ + z(v,,))_> bsllv, + z(v.)ll~, J
-PQ
Q
by (hi), (z(v~)) is b o u n d e d in L '~. Thus z ( v ~ ) - r in L " a l o n g a subseguence. Since
~QH(X, t,
~'n
+ zG')) >- f H(x, t, )
Q
1,'n
+ z(v,,)),
We have
[oH(X,t, V+-(-~)) > lim f o H ( x , t , vn + >_ lira f H(x, t, v,, + z(v~)) t ? ~o o
JQ
>_ [ H ( x , t , ~ + 2 ) Q by the weakly lower semlcontinuity, and so 3 = z(~) by the minimality, and
lim IQH(x,t, vn + z(v,,)) = f Q H ( x , t , ~ + z(~)).
n~oo
N o w by L e m m a 2.3, noting that v~ + z(v,,) - T + z(T).
I H(x, t. (v,
+
z(v,,)) - (v+
2(i7)))
0.
O and then by Remark 2.1,
H(v,, - ~) + (z(v,) - z(i:))ll~ - 0 which implies (since vn - ;;) z(vn) - z(~), proving (ii). 3. T H E S T R I C T L Y I N C R E A S I N G C A S E
In this section we prove Theorem 1.1. T h r o u g h o u t the section we assume that f satisfies (fl). Let x be the Banach space given by (2.1) with the n o r m (2.3) and the direct sum decomposition (2.2). Set X ~ =Xff, X 2 = X ~ X ~ ~N~ (3.1)
Periodic solutions of a superlinear wave equation
271
if case (1) or case (2) o f Theorem 1.1 occurs, and
);fl = x f ~ X 0,
X2 = X b ~ Ne~
(3.2)
if case (3) o f Theorem 1.1 occurs. Then
X = X l ~ X 2. Suppose that x, l c
c
. . . c x ',
x?
c
c
. . . c x e
are two sequences o f finite dimensional subspaces such that
X j = [..JX~, j = l , 2 . nEN
In what follows, two multiindices 0 = (0 I, 0 e) and r = (V 1, -r e) E N z are denoted by 0 _< r if 01 _< r l, 02 _< r e, and a sequence (0,) c N e is said to be admissible if for each 0 E N ~ there exists m ~ N such that 0n >-- 0 whenever n >__m. For 0 E N 2, let xo
= xJ,
xge.
For a functional • 6 C1(X, R) we denote by ~0 = ~lxo the restriction on Xo. Recall that • is said to satisfy the condition (PS)* if every sequence (uo.) with (0.) being admissible such that t
uo. E Xo., s u p S ( u 0 . ) < oo and cbo.(uo .) - 0 as n - 0o n
contains a subsequence converging to a critical point o f ~. In addition, • is said to satisfy the local linking condition at 0 if for some p > 0 • (u) >_ 0 whenever u E X 1 with [lull - p, • (u) _< 0 whenever u 6 X 2 with Hull < p. The original local linking condition and a r g u m e n t was introduced in [6] under a stronger assumption. Silva in [7,8], Brezis a n d Nirenberg in [9] have used this improved version. Clearly, in this case, 0 is a trivial critical point o f ~. Recall that a function u E X n L °° is a weak solution o f (W) if for all (io ~ X,
~Q[uXCPx - utcPt - f ( x , t, u)qg] = 0. Formally the integrand for
I x2 - u~2 _ F ( x , t , u ) ) ] • (u) =- ~Q [~(u is the Lagrangian for our problem. However it seems to be difficulty for us to treat • as a functional on X since there is no upper b o u n d on the growth o f f as lu[ ~ co. Hence we need to modify f . For k E N with k >_ ? (? is the constant in (f3)), set
f~(x,t,~)=
f ( x , t, ~) min{f(x,t,g),l+
f(x,t,k)+Rk(~ a-l-ka-l)} m i n { f ( x , t, g), - 1 + f ( x , t, - k ) + Rk(k "-L - ( _ ~ ) a - l ) }
for 1~51--- k, for ~ > k , for ~ < - k ,
272
DING YANHENG and L1 SHUJIE
Rk > 0 to be determined. Let gk(x,t,~)
= fk(x,t,~)-
b~,Fk(x,t,~)
= ~ o f k ( x , t , s ) ds and
Gk(x, t, ~) = ~o~gk(x, t, s) ds. Clearly Ifkl-< [fl
for all k,
(3.3)
and (fs) Ifk(x, t, ~)1 -< b ll~l + b2l~l =-1 for all x, t, ~, where b2 depends on k. It is also clear that fk satisfies (fl), (f2), (f4). For Rk first we require it to satisfy Rk >- max
{l+f(x,t,k) k,_l
1-f(x,t,-k)} k,_l
for all (x. t) c Q.
An explict computation shows that fk satisfies (f~) with the same constant ?. We then further require Rk to be chosen so that Ifk-x[-< Ifkl
forall k.
(3.4)
To see this, inductively, suppose we have already found Rk t- By definition,
fk(x, t, ~) = f ( x , t, ~) > fk(x, t, ~) whenever k _< ~ < k + Yk with some Yk > 0 small and independent of Rk. Let now Rk be such that
f ( x , t, k) + Rk(~ ~-t - k c'-I ) > f ( x , t, k - 1) + Rk-l ( ~ - l
_ (k - 1) a-1 )
for all ~ >_ k + Yk. With this choice of Rk one can easily verify that fk >- fk-1 for ~ _> 0. A similar argument for ~ < 0 gives (3.4). Finally, by (f~)-(f4), it is not difficult to see that there are positive constants b3 and b4 both are independent of k, (since ? does not depend on k and we only consider b > 0) such that
fk(x, t, ~)~ > aFk(X, t, ~) - b3,
V(x, t, ~)
(3.5)
and
Fk(x,t,~) >--b41~l ~,
V(x,t,~).
(3.6)
We consider the functional dPk(U ) = ~l fQ(Ig2x -- 122) - f O Fk(x,t,U) l
-
2-bu2) = 2 QI 1 = ~Q(u) - ~g(u)
J Q Gk(x't'u)
where Q(u) = 5Q(U2 - u~ - bu2). It is well known, under (f~)-(fs), that ~k ~ C ~(X, R) and each critical point uk of % is a (weak) solution for
Du = fk(x, t, u).
(3.7)
Moreover, Uk is also a solution of (W) if [[uklloo < k. Therefore there are two things to be done: first, to look for nontrivial critical points of ~k, and then to establish some bounds, independent of k for k large, on the critical points. LEMMA 3.1. ~k satisfies the (PS)*.
Proof. For notational convenience we denote (P = qbk in what follows. We will also denote by dg different positive constants.
Periodic solutions of a superlinear wave equation
273
Suppose uo. ~ Xo., sup, ~(u0.) _< dl and ¢b'.(uo.) = ¢b'o.(uo .) ~ O. Then by (3.5)-(3.6), for n large, writing u = uo., dl + Llull --- ~ ( u ) - ~t (~'n(U), U) _> d2 f o F k ( X , t, U) - d3 > d4llullg
-
d3.
and so ds(1 + Ilullt/% Since dim X° < oo, all norms on X° are equivalent, we see by (3.8) ilullo,
(3.8)
-<
Ilu°ll _< d6(1 + Ilull~/%.
(3.9)
Since Na is closed in L a with a closed complement L~ -- clL. (X~- • X ° • X ~ ) (where ClL. (S) denotes the closure of the set S in L~), the projector along L~' onto No, is bounded, and it follows from (3.8) that Ilzll. -< d7(l + Ilu111/%. (3.10) Noting that Q(u) is positive definite on X~ and I1( Q ' ( U) , U+ ) = (¢b'n(u), U+ ) + ] 0 g k ( x , t , u ) u +
we get IIu + IIz _< ds IIu + II + d9 Jo Igk(x, t, U) II U+I which, jointly with (f5), yields (o( = c¢/(c¢- 1)) Ilu÷ll2 -< dsllu+ II + dl0
(fo
(lul + lul ~-~)
Ilu+ll~
-< all(1 + Ilull~-l)llu+ll. Combining this and (3.8) implies Ilu+ll ~ d12(l + ltull<~-l/:%.
(3.11)
Ilu-lL ~ d13(1 + Ilull<~-l>:~).
(3.12)
Similarly we also have It then follows from (3.9)-(3.12) that Ilull ~ d14(1 + ilull~-l)/%. Hence (uo.) is bounded. Recall that X~ are compactly embedded in L ~ and dim X° < co. By the boundedness of (uo,), one can assume that u ° ~ -° u in X, vo, - Uo-" + u ° + u~, ~ -~ in X , vo, - -~ in L ~, and zo, - -g in L ~. Since Xb+- and X° are Hilbert spaces and N~ has a Schauder basis, we may assume that the subspace sequence (Xo.) has been chosen so that the associated projection operator sequence (Po.) is bounded, i.e. II1'o. II <- const., and (I - Po,)u - 0 for each u. Since ~',(uo,) - O, zo, = Po, zo, and, according to Lemma 2.4 (recall that f k satisfies all the assumptions of the lemma), z(vo.) ~ z(V), f Q f k ( x , t, I:O, + ZOn) (Zo. - Z(~on) ) = ( ~ ' ( u o . ) . ZO. - z(vo.)) = (¢b'n(uo.),zo. - z ( v o . ) )
- (~b'(uo.), ( I - P o . ) z ( v o . ) )
~ O.
274
D I N G Y A N H E N G a n d LI S H U J I E
Hence passing to the limit in the inequality
IQFk(x't'v°"+z(v°'))-~QFk(X't'v°"+z°°)>f
fk(x't'v°'+z°")(z(v°')-z°")Q
one o b t a i n s
I
Q
Fk(x,t,f+z(f))
> lim
n-e~
Q
& ( x , t , vo,+zo,,)
-> ~olimf Q & ( x , t, vo, + zo,)
>__~QFk(x,t,-~ + 2) > j'QFk(X, t,-~+ z(7)) a n d so ~ = z(~) a n d
IQFk(x,t, vo, + zo,) -- ~QFk(x,t, v + z(v)). N o w a p p l y i n g L e m m a 2.3 yields I1% +zo, - 5 + z(5)[[~ - 0 a n d zo° ~ ~ = z(V). Therefore uo° ~ V + z(V) in L ~, a n d a s t a n d a r d a r g u m e n t shows that W(uo,) - W ( ~ ) = ' g ' ( 7 + z(V)). Since O',(uo,) ~ O, it follows that Q'(uo,)/2 ~ W(K) which implies u~, ~ ~± a n d so uo, ~ ~ in X . Finally, it is easy to see that ~ is a critical p o i n t o f qs, a n d the conclusion follows. LEMMA 3.2. qSk satisfies the local linking c o n d i t i o n at 0.
Proof. First a s s u m e case (1) o r (2) o f T h e o r e m 1.1 occurs. M o r e o v e r we consider only case (2) since case (1) can be treated similarly. Recall that in this case X 1 = Xb+ given by (3.1). N o t i n g t h a t Q(u) is positive definite on X 1, we have 1Q(u) > dlHul] 2,
V u E X 1.
(3.13)
By (f4) Gk(x,t,~) = o(l~L 2) as ~ ~ 0 u n i f o r m l y in (x,t) ~ Q, a n d so for any E > 0 there is d~ > 0 such t h a t [Gk(x, t, ~)[ < E~ 2 + d~l~[ ~ which implies t h a t [~(u)[ < Ed21[ul[2 + d31lul[ ~, or 't'(u) = o ( l l u f )
as
u ~ 0
(3.14)
It then follows f r o m (3.13)-(3.14) t h a t there is a p > 0 so t h a t I rb(u) = rbk(u) >_ ~dlllul[ 2 _> 0 V u E X l C~Bp
(3.15)
rb(u) <_ O, V u ~ X 2 = X ° • Xd- • Nc, with Ilull -< p
(3.16)
N o w we prove p r o v i d e d p being small e n o u g h . T h e a r g u m e n t a n a l o g o u s to t h a t o f [2, P r o p o s i t i o n 3.1]. F o r the reader's convenience, we sketch the proof. Let u = u ° + u - + z ~ XZ • Xb- • N~ a n d C = s u p { l , ( u ) "u ~ X 2 n B o } .
Periodic solutions of a superlinear wave equation
275
Since ~ l x ~- is weakly u p p e r semicontinuous. So the s u p r e m u m is attained at some ~ E Bp. If Ilu-II = p and p is small enough then one immediately get
cI,(u)<_O-~ep t
2
<0
VuE
X
2
n B- p
for some E > 0. Hence Ilff-II < P. This implies that t~(P^u") = . P ^ A ( x , t,
in the sense o f distributions where A = {(j, k) E N x Z ; j 2 - k 2 < 0}. F r o m L e m m a s 2.1 and 2.2 we get that I1~11oo -< I1~°11 + lib'- Iloo + IlYtloo -< 6 i f p is small enough, where 6 was given in T h e o r e m 1.1. Therefore from that G(x, t, ~) >_ 0 as I~1 -< 6 we get ¢b(~ <_ - ~ Q G ( t , x , u ~ d x d t
<_ O
this proves (3.16). In the case (1) we do not require that G(x, t, ~) >_ 0 as I~1 -< 6 because @(u") ___ -dallH- 112/2 - bll~ll~/2 + e(p)llHIl~ -< 0 as p small• In case (3) X 1 = Xb+ • Xb~, X 2 = Xb- • N , and let C = inf{~(u) • u E X l n Bo}. An argument analogous to that showing (3•16) gives ~ ( u ) >_ 0 V u E X 1 n Bp. For u E X 2 consider sup{~(u) • u E X 2 n Bp} and the p r o o f is similar to case (2). P r o o f o f Theorem 1.1. If for each k there is a critical poiut uk u: 0 0 f @ k such that ~k(Uk) < c where c is independent o f k then it is easy to prove that Ilukll~ is b o u n d e d and therefore Uk ~ 0 is a solution of (W) if k is large enough. (a) Suppose that there is a k0 such that Yp - @k0 has no critical point u ~ 0 with @(u) _< 0. By the argument to that [10, T h e o r e m 2] we can define a m a p p i n g y • aQ~ - x,,~1 +1 ~ )(2 such that ~(y(s, u)) _< 0 for all (s, u) E ~Q, where Q, = [0, 1] x (X 2 n B p ) and ml > 0 is a positive integer. F r o m Ifk01 < Ifkl as k >_ k0 it follows that ~k(y(s, u)) _< 0 for all (s, u) E OQ, as k >_ k0. Let F = {~, e C ( Q , , xtj) " ~I~Q, = z} where/3 = (n, n) with n _> ml + 1. Choose Pk SO that ~k > 0 on X l n Bp,. It has been shown in [10, T h e o r e m 2] that for each sufficiently large n one m a y deform X2 n Spa to a set Sk C X2 such that @k >-- Ck > 0 on Sk where Ck is independent o f n and the sets ¥(aQn) and Sk link nontrivially. Let G ( k ) = inf
sup
~k()'(s,u))
~EF (s,u)EQ,,
Since "~ 1 I C -= sup{@(u) " u E X,~,+ 1 • X 2 } >- sup{dPk(U) " u E A"~, +1 • X 2 }
therefore Ck <- Cn (k) <_ C where C is independent o f k. A s t a n d a r d a r g u m e n t show that C,(k) is a critical value of (~k)a where/3 = (n, n). Using (PS)* condition and letting n - oo we get a critical poiut Uk of CI'k with 0 < Ck <- ~k(Uk) <-- C. It implies that Uk ~ O, Ilukll~ is b o u n d e d and Uk is a solution o f • as k is large enough. (b) Suppose now that k0 stated in (a) does not exist. Then for each k there is a critical point Uk ~ 0 of @k with @k(Uk) --< 0. This implies Uk ~ 0 is a solution o f (W) if k is large enough. T h e p r o o f is complete. •
276
D I N G Y A N H E N G and LI SHUJIE 4. THE N O N D E C R E A S I N G CASE
F r o m now on we always suppose that all the assumptions of T h e o r e m 1.2 are satisfied. Recall that we are only interested in the case b > O. LEMMA 4.1. For each k >_ ?, there are Pl > 0, A > 0 depending on k, and P2 > 0 independent of k, such that (~k(U)>_O V uEXINBp~, ~k(U)>~ V uEXIAOBp~, (4.1)
• k(U) --< 0 V u C X 2 n Bp2-
(4.2)
Proof. Recall that if case (2) of T h e o r e m 1.2 occurs, then X l = X~+ and X 2 = X~ • Xff • Nc~. (4.1) is clear by the first part of the p r o o f in L e m m a 3.2. We consider (4.2). Since g(x, t, ~)E > 0 whenever IEI <- 6, Ifk(x,t,E)] > ]bEI iflE[ -< 6. Recall that for IE] >- E fk(x, t, ~)E >- Fk(X, t, ~) >_ dl~l ~, and so fk(x, t, ~) >_ d~ ~-1 for ]El > ?, and fk(x, t, E) <- - d ( - ~ ) ~-1 for ~ _< -?. Let .f(~) ~ C(R, R) be a strictly monotonically increasing function given by aT(~ ) =
and f ( ~ ) = - f ( - ~ )
[
b~ ll(E) 12(~) dEC,-1
if 0 _< E -< 6/2, if 6 / 2 < E <~, ifff
for ~ < 0, where
if-
b6 +~. 2d(2~)c,-2
ll(E) is a linear function determined by /l(~i/2) = b(6/2), 11(7~) = b6, and /2(E) is a linear function determined b y / 2 ( x ) = b6,/2(27) = d(27) ~ i Then Ifk(x, t, E)I > [Y(~)I, and
& ( x , t, ~) >_ IF(E) = I ~ f ( s ) ds,
for all
~ - ~O F ( ~ ) . Then Let g)(u) = ~l ~o(U;.~ - UT) cI,~(u) _< gP(u)
for all u E X.
By the second part of the p r o o f for case (2) in L e m m a 3,2, there is a P2 > 0 such that g~(u)_<0,
V u~X
2nBp:
and (4.2) is proved in this case. If case (3) of T h e o r e m 1.2 occurs, then X I = Xh+ • X~, and X 2 = X~- • Na. Set fk(x, t, ~) = fk(x, t, ~) + A2o¢I~I~-2E. Then
-~k(x, t, E) = Fk(X, t, ~) + A2{~I ~ --= ~E 2 + (Gk(x, t, E) + A21EI% > Fk(X, t, E),
'I e (u2-u~)-I o i~(x' t, u)
g)k(U) = ~
= ~(u)
- A211ullg -< ~ ( u ) .
Periodic solutions of a superlinear wave equation
277
Since ~ ( x , t, ~) is strictly increasing in ~, and (Gk(x, t, ~) + A21~I") -< 0, by the first part of the p r o o f for case (3) in Lemma 3.2, there is a Pl > 0 such that • k(U)>-YOk(U)>O
V ueX
1c~,.
We show that ¢bk[Xt~S~ >_ A for a A > 0. If not, there are u~ ~ X ~, ][u~ll = p~ such that
0 <_ ~k(U.) = ~k(u.) + A2llu, ll~ ~ 0. Then I[u~ [I ~ 0 and so HUn + I[ - P~. We see that ]
+
[
'l'k(U.) = ~Q(u. ) - j >- ~ Q(u.+ )
-
+
Q
Gk(X, t, u. + u~)
Cl Ilu.[l~
c211u.ll~.
the left-hand side tends to 0. while the right-hand side tends to a positive constant, a contradiction, (4.1) follows. Next, to prove (4.2), we again define a function f ( ~ ) ~ C(R, R) by
=
and f ( ~ ) = - f ( - ~ )
b ~ - h l p ~ °-I
for 0 _ ~ _< 6/2
Ii (~) /2(~) d~ ~-1
for 6 / 2 < ~ <_ for 2 < ~ < 2? for ~j ___2?
for ~ < 0, where
6(b - pal (~)p-2) X=
+?,
4d(2?)a-:
II (~) is a linear function determined by Ii (6/2) = (b - pat (6/2) p-2) ~, Ii (-£) = ( b - pAI6p-2)6, and l:(~) is a linear function determined by 12(~) = II (~), 19(27) = d(27) " - l . Then it is easy to verify that f ( ~ ) is strictly increasing, fk(x, t, ~) = b~ + gk(x, t, ~) >--f ( ~ ) , and
Fk(X, t, ~) >__ff(~) =ff(~) =
~2 + ( - h l [ ~ f )
f ( s ) ds, if I~1 <-
~.
By the second part of the p r o o f for case (3) in L e m m a 3.2, one sees that there is a P2 > 0 such that
• k(u) <-~(u) - 1 fQ(u2_u2t) - f Q f f ( u ) < O , The lemma is proved.
V uEX2n-Bm.
•
N o t e that since Ifk-I (x, t, ~)1 --< Ifk(x, t, ~)1, without loss of generality, we can assume that A(k) _< ~k(u) <- A,
V u E X t n OBp,
(4.3)
In this section the notations X, X~, X~,, X b, N~, Xn, Xo, Po still have same meaning as before. Since 79 - { s i n j x s i n k t , s i n j x c o s k t ; (j, k) E N × Z} is a Schauder hasis for X,
IIPo.[l£Ix,x) <- Mi
(4.4)
278
DING YANHENG and LI SHUJIE
for some M1 > 0, and ]l(l-P0,,)u[]
-0
as
n-
~
(4.5)
for each u ~ X. Noting that, for any q ~ (1, ¢~), :P is also a basis sequence in Lu which can be extended to a Schauder basis for LU Po, can be extended as a projector from L u onto Xo,, such that IIPe~IIz
P r o o f For notational convenience, write ff = ~k, 4% = (cbk)o,,, and let d, denote positive constants, which are independent of k, and c(k) denote positive constants depending only on k and varying with the places. Since ~b(uo,) _< A + 1 and cb'~(uo,) = 0. (4.7) we have
A + 1 > ¢b,(uo,) - 2i (~'.(uo,,), uo°) f = Jo{½fk(x, t, uO,)uo, -- Fk(x, t, UO°)] which, together with (3.5)-(3.6), implies
IO Recall that, f o r u E X ,
Fk(x't'u°")<-~Q fk(x't'u°")u°°<-d'
(4.8)
tlu0,ll~ -< d2.
(4.9)
u = u + + u ° + u- + z - v + z. By (4.7),
Q(u~ ) = (~I Q, (uo,),uo,> + = fQ g k ( x , t , uo,)u~. +
(4.10)
Since Q(u) Is positive definite in X# and negatine deginite in X # , (4.10) and (J~) as well as (4.9) yield Ilu~,,ll -< c(k)llu~o[l~,. (4.11) Noting that dim Xh° < oo by (4.9) Ilu~,ll -< d31lu~,l], -< da[luo°ll,~ <- ds.
(4.12)
Since Na is a closed subspace of L " with the closed c o m p l e m e n t clL~ ( X ) , the closure of X in L ~, the projector from L " onto Na is bounded, and so IIz0, ll -- Ilzooll~ -< d611uooll~, <- d7 by (4.9). N o w (4.1 1)-(4.13) show that Iluo, II ~ c(k).
(4.13)
Periodic solutions of a superlinear wave equation
279
Therefore, without loss of generality, one can assume
uo,-~k
in X,
vo, ~-fk in L ~.
(4.14)
In virtue of (4.7)
Duo. = Po, fk(x, t, uo,)
(4.15)
and consequently, it follows from (4.6), (fs) and (4.9) that IIDu0.11., = NPo.fk(x, t, uo.)ll., _< M . , Ilfk(x, t, uo.)ll.' -< c(k). where a ' = a / ( a - 1). Hence one can assume Duo. - ~ in L ~'. Since, by (4.14), Duo. ~ D-~k in the sense of distributions, g = t3~k. N o t e that for each qo E X,
I
Q
+ fQ(fk(x, t, uo,)(I - Po,)cP
(4. 16)
>-- j (fk(x,t, uo,)(I-Po,)q). Q Since I1(I - Po.)q)ll - 0 (see (4.5)) and by (4.14)
IQ °u°" " u°" = IQ rTu°" " v°" -- ~Q D ~ " ~k = ~ r T ~ k ' ~ passing to the limit in (4.16) yields
O(O~k - f k ( x , t, qo))(Kk - qg) >__0.
(4.17)
Inserting in (4.17) the qo = uk + sx, where s > 0 and X E X, dividing by s and letting s ~ 0 give Q(D-iik -- f k ( x ,
t, -ffk))X >-- O.
(4.18)
Since X was chosen arbitrarily, (4.18) shows that ~k is a weak solution for
D-ak = fk(x, t, ~k).
(4. 19)
N o w we show that Kk ~: 0. Arguing indirectly, assume Kk = 0. Then yD, ~ 0 in L a, which, jointly with (4.11), implies vo, ~ 0 in X, and
dlluo, ll~, <- ~QFk(x,t, uo,) <- IQfk(x,t, uo,)Uo, = fQmuo, • YD, <<-c(k)llvo, ll~ -- O. Therefore IIz0.11a ~ 0, and then we get 0 < p _< Iluo.II -< Ilvo.II + Ilzo. II - 0.
a contradiction.
280
D I N G Y A N H E N G and L1 S H U J I E
Next we establish bounds for Iluktlo~. Note that, by (4.19)-(4.15),
Jef*(x't'~*)~*=Je
(m~,-=uo,,)v,+fe
=,oo(V,-vo,)+fef,(x,t,,o,,),o
(4.20)
Since the first and second terms of the right-hand side of (4.20) tend to 0 (as n ~ ~ ) , by (4.8).
IQf,
(x. t.~,)~- <_ d~,
and so
>1
fk(x, t, K~)-Kk + I
]~_<1
IJ~(x, t, ~ ) 1 _< ds.
Hence, by (4.19) according to [11] or [1], II~k[Io~ ~ d9llfk(x,t,~k)lll ~ dlo. Since fk satisfies (f~), for each R > 0, there is dR > 0 depending on R but independent of k such that ]fk(x, t, g + ~) - fk(x, t, ~)l -> dRlg[ - 1 whenever Igl -< R. Taking into account [2, R e m a r k 3.2] with R = dl0, we get
Ilrkll~ -< dll where ~k = ~k + ~ . Hence I1~11~ -< d12. Therefore for k large, say, k >_ d ~ + 1, ~ is a nontrivial weak solution of (W). The lemma is thereby proved• •
Remark 4.1. In virtue of L e m m a 4.2, to prove T h e o r e m 1.2, it remains to consider the case that there is a k0 >- ~ such that, for every p > 0 and each k >_ k0, there exists 0% = ocp(k) = (mp, mp) ~ N 2 satisfying, for any o¢ >_ ocp (~k)'~(u) ~e0,
V uEX~
with ]lull >-p and ~ k ( u ) - < A +
1.
Proof of Theorem 1.2. Let ~ = ~k0 (see R e m a r k 4.1). Let P2 be the constant in L e m m a 4.1. By R e m a r k 4.1, there is mp2 = mo2(k) such that for ¢x >_ O~p2 = (mp2, mp:), ~'c~(U) "~ O, Vu c Xc, with Ilul[ -> P2 and ~ ( u ) _< A + 1. It follows from (f~) that i f R is large enough, then ~ ( u ) _< 0 for each u ~ X m,2+l ~ @ X 2 with I[u[[ >- R/~/~. Let e~ = (mp~_,n), n >_ mp:+l. Since dim Xc,
Q. = [0, 1 ] x ( x 2 n ~p: ) and define a m a p p i n g y • 3Q~ - X , oo~+~X~by 1 f o r 5" = 0
y(s, u) =
o-(2s, u) 2(1 - s)cr(1, u) + ( 2 s - l)v0 V0
b/ C 2 2 f-'l Bp2,
fors~(0,½] 1
for s e (~, 1) for s = 1
u~X }n&2, u E X} n Sp2, ucX 2nBp2.
Periodic solutions of a superlinear wave equation
281
Then ~(¥(s, u)) __ 0 for all (s, u) E OQn. Since Ifk01 --- Ifkl i f k _> k0, it follows that ~k(y(s, u)) __0 for all (s, u) E ~Qn whenever k >_ k0. Let/3 = (n, n) whenever n > mp2 + 1, Yl~e.= Y}
F = {~ E C ( Q . , X o ;
and cn(k) = inf
sup r~k(~'(s, u)),
Vk >_/co.
Let p~ = Pl (k) be the constant in Lemma 4.1 so that • k(u)->0 on X l n B p , ,
~ k ( u ) > - - A = A ( k ) on X lnSp~.
By Remark 4.1, for all a >_ ap~(k), ('bk)'~(u) * 0 for u ~ X~, Ilull >- Pl and '~k(U) <-- A + 1, particularly, for all u E X1 n So, with a = (n, n) >__apl (k). Therefore we may deform X.l n So, to a set S c X., using a pseudo-gradient flow for (~k)~, such that ~k(U) >- A ( k )
V u E S,
(see [10, Lemma 1]). Moreover, along a line in the proof of [10, Theorem l, step (3)], one can show that y(aQ,) and S link (i.e. ~(Qn) n s * qb for any ~ E F). Therefore C.(k) > A(k). 1 1 In addition, as y(0Q.) c X,.~2+l • X2, there exists a y0 c F with y0(Q.) c X,~.2+l • X2. Since 1 d = s u p { ~"~( u ) ; u E X.rap2 l + I ~ X2} > sup{d~k(U);U E Xmp2+l O X'2},
we have C.(k) <_ d
(d independent of k). A standard argument shows that C.(k) is a critical value of (CI'k)t~(/3 = (n, n)). Let uo(k) ~ XO be the associated critical point of (~k)~. Then A(k) __<~k(U~) --< d,
(~k)'(u~) = 0.
Repeating the procedure for proving Lemma 4.2, one sees there is a weak solution fik for t3~k = f k ( x , t, -Ok).
and II~kll~o --< dl independent of k. Now we show that ~k :~ 0. If not, one can verify that Ilu~ll- 0 in X. Then we obtain that A(k) _< '~k(U~) -- 0 as /3 ~ 0% a contradiction. Finally, since II~kllo~ -< d~ independent of k, for k large, Uk is a nontrivial weak solution of (W) The proof is thereby complete. •
282
D I N G Y A N H E N G and LI S H U J I E REFERENCES
I. R A B 1 N O W I T / . t: 1t., Free vibrations tor a semilinear wave equation. Comm. Pure Appl. Math., 1978, 31, 31 68. 2. LI. S. J. & SZI ,41N A . Periodic solution for a class of n o n a u t o n o m o u s wave equations differential and integral equations, to be i ' ~,i bed. 3. W I L L E M , I~!. ,, ~ h ~monic oscillations of a semilinear wave equation. Nonlinear Analysis, 1985, 9, 503 514. 4. C H A N G , K. * .~ ' ~J. J. Q., The existence of infinitely m a n y solutions lbr the semilinear wave equation, to be published. 5. D I N G . Y. ,L, A l, ' on nonlinear beam equations (in Chinese) Acta Math Sinica. 1990, 33, 172 181. 6. LIU. J. Q. & I I Some existence theorems on multiple critical points and their applications. Kexue Tongbao, 1984. 17. 7. SILVA. E. ~,. B ~r:I'cal point theorems and applications. P h D. thesis, Univ. of W i s c o n s i n - - M a d i s o n , 1988. 8 SILVA. E. A. B r - :ng theorems and applications to semilinear elliptic problems at resonance. Nonlinear Analysis, 1991. 16. 455 -* ' 9. B R E Z I S . It. & " ' . N B E R G , L.. Remarks on finding critical points. Comm. Pure Appl. Math., 1991.64, 939-963. 10. LI. S. I & '~\ l[.J .,i. M.. Applications o f local linking to critical point theory. J Math. Anal. Applie., 1995, 189, 6-32. 11. BREZI'~. it.. COtv~ ': J. M. & N I R E N B E R G , L.. Free vibrations tor a nonlinear wave equation and a theorem of P. Rd~'!;~owJt7 e ,.,n Pure dppl Math., 1980.33, 667 689